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Full text of "Aristarchus of Samos, the ancient Copernicus; a history of Greek astronomy to Aristarchus, together with Aristarchus's Treatise on the sizes and distances of the sun and moon"

ARISTARCHUS OF SAMOS 

^j THE ANCIENT COPERNICUS 

A HISTORY OF GREEK ASTRONOMY TO ARISTARCHUS 

TOGETHER WITH ARISTARCHUS'S TREATISE 

ON THE SIZES AND DISTANCES 

OF THE SUN AND MOON 

A NEW GREEK TEXT WITH TRANSLATION 

AND NOTES 



BY 



SIR THOMAS HEATH 

K.C.B., Sc.D., F.R.S. 

SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE 



150471 



OXFORD 
AT THE CLARENDON PRESS 



■ 



•• • • » 

* * ' » i • 



HENRY FROWDE, M.A. 

PUBLISHER TO THE UNIVERSITY OF OXFORD 

LONDON, EDINBURGH, NEW YORK, TORONTO 

MELBOURNE AND BOMBAY 






PREFACE 

This work owes its inception to a desire expressed to me by 
my old schoolfellow Professor Η. H. Turner for a translation of 
Aristarchus's extant work On the sizes and distances of the Sun and 
Moon. Incidentally Professor Turner asked whether any light 
could be thrown on the grossly excessive estimate of 2° for the 
angular diameter of the sun and moon which is one of the funda- 
mental assumptions at the beginning of the book. I remembered 
that Archimedes distinctly says in his Psammites or Sand-reckoner 
that Aristarchus was the first to discover that the apparent diameter 
of the sun is about 1/7 20th part of the complete circle described 
by it in the daily rotation, or, in other words, that the angular 
diameter is about ^°, which is very near the truth. The difference 
suggested that the treatise of Aristarchus which we possess was 
an early work ; but it was still necessary to search the history of 
Greek astronomy for any estimates by older astronomers that 
might be on record, with a view to tracing, if possible, the origin 
of the figure of 2°. 

Again, our treatise does not contain any suggestion of any but 
the geocentric view of the universe, whereas Archimedes tells us 
that Aristarchus wrote a book of hypotheses, one of which was 
that the sun and the fixed stars remain unmoved and that the 
earth revolves round the sun in the circumference of a circle. 
Now Archimedes was a younger contemporary of Aristarchus; 
he must have seen the book of hypotheses in question, and we 
could have no better evidence for attributing to Aristarchus the 
first enunciation of the Copernican hypothesis. The matter might 
have rested there but for the fact that in recent years (1898) 
Schiaparelli, an authority always to be mentioned with profound 
respect, has maintained that it was not after all Aristarchus, but 
Heraclides of Pontus, who first put forward the heliocentric 



IV 



PREFACE 



hypothesis. Schiaparelli, whose two papers Le sfere omocentriche 
di Eudosso, di Callippo e di Aristotele and I precursor i di Copernico 
nelV antichita are classics, showed in the latter paper that Heraclides 
discovered that the planets Venus and Mercury revolve round the 
sun, like satellites, as well as that the earth rotates about its own 
axis in about twenty-four hours. In his later paper of 1898 (Origine 
del sistema planetario eliocentrico presso i Greet) Schiaparelli went 
further and suggested that Heraclides must have arrived at the 
same conclusion about the superior planets as about Venus and 
Mercury, and would therefore hold that all alike revolved round 
the sun, while the sun with the planets moving in their orbits 
about it revolved bodily round the earth as centre in a year ; 
in other words, according to Schiaparelli, Heraclides was probably 
the inventor of the system known as that of Tycho Brahe, or was 
acquainted with it and adopted it if it was invented by some 
contemporary and not by himself. So far it may be admitted 
that Schiaparelli has made out a plausible case ; but when, in the 
same paper, he goes further and credits Heraclides with having 
originated the Copernican hypothesis also, he takes up much more 
doubtful ground. At the same time it was clear that his argu- 
ments were entitled to the most careful consideration, and this 
again necessitated research in the earlier history of Greek astronomy 
with the view of tracing every step in the progress towards the 
true Copernican theory. The first to substitute another centre for 
the earth in the celestial system were the Pythagoreans, who made 
the earth, like the sun, moon, and planets, revolve round the central 
fire ; and, when once my study of the subject had been carried 
back so far, it seemed to me that the most fitting introduction to 
Aristarchus would be a sketch of the whole history of Greek 
astronomy up to his time. As regards the newest claim made by 
Schiaparelli on behalf of Heraclides of Pontus, I hope I have 
shown that the case is not made out, and that there is still no 
reason to doubt the unanimous testimony of antiquity that 
Aristarchus was the real originator of the Copernican hypothesis. 
In the century following Copernicus no doubt was felt as to 



PREFACE ν 

identifying Aristarchus with the latter hypothesis. Libert Fro- 
mond, Professor of Theology at the University of Louvain, who 
tried to refute it, called his work Anti- Aristarchus (Antwerp, 
1 631). In 1644 Roberval took up the cudgels for Copernicus in a 
book the full title of which is Aristarchi Samii de mundi systemate 
partibus et motibus eiusdem libellus. Adiectae sunt AL+ P. de 
Roberval) Mathem. Scient. in Collegio Regio Franciae Professoris, 
notae in eundem libellum. It does not appear that experts were 
ever deceived by this title, although Baillet (Jugemens des Savons) 
complained of such disguises and would have had Roberval call his 
work Aristarchus G alius, 'the French Aristarchus,' after the 
manner of Vieta's Apollonius Gallus and Snellius's Eratosthenes 
Batavus. But there was every excuse for Roberval. The times 
were dangerous. Only eleven years before seven Cardinals had 
forced Galilei to abjure his * errors and heresies ' ; what wonder 
then that Roberval should take the precaution of publishing his 
views under another name? 

Voltaire, as is well known, went sadly wrong over Aristarchus 
(Dictionnaire Ρ hilosophique > s. v. ' Systeme '). He said that Ari- 
starchus ' is so obscure that Wallis was obliged to annotate him 
from one end to the other, in the effort to make him intelligible ', 
and further that it was very doubtful whether the book attributed 
to Aristarchus was really by him. Voltaire (misled, it is true, by 
a wrong reading in a passage of Plutarch, De facie in orbe lunae, 
c. 6) goes on to question whether Aristarchus had ever propounded 
the heliocentric hypothesis ; and it is clear that the treatise which 
he regarded as suspect was Roberval's book, and that he confused 
this with the genuine work edited by Wallis. Nor could he have 
looked at the latter treatise in any but a very superficial way, or 
he would have seen that it is not in the least obscure, and that the 
commentary of Wallis is no more elaborate than would ordinarily 
be expected of an editor bringing out for the first time, with the 
aid of MSS. not of the best, a Greek text and translation of a 
mathematical treatise in which a number of geometrical propositions 
are assumed without proof and therefore require some elucidation. 



IV 



PREFACE 



hypothesis. Schiaparelli, whose two papers Le sfere omocentriche 
di Eudosso, di Callippo e di Aristotele and I precursor i di Copernico 
nelV antichita are classics, showed in the latter paper that Heraclides 
discovered that the planets Venus and Mercury revolve round the 
sun, like satellites, as well as that the earth rotates about its own 
axis in about twenty-four hours. In his later paper of 1898 (Origine 
del sistema planetario eliocentrico presso i Greet) Schiaparelli went 
further and suggested that Heraclides must have arrived at the 
same conclusion about the superior planets as about Venus and 
Mercury, and would therefore hold that all alike revolved round 
the sun, while the sun with the planets moving in their orbits 
about it revolved bodily round the earth as centre in a year ; 
in other words, according to Schiaparelli, Heraclides was probably 
the inventor of the system known as that of Tycho Brahe, or was 
acquainted with it and adopted it if it was invented by some 
contemporary and not by himself. So far it may be admitted 
that Schiaparelli has made out a plausible case ; but when, in the 
same paper, he goes further and credits Heraclides with having 
originated the Copernican hypothesis also, he takes up much more 
doubtful ground. At the same time it was clear that his argu- 
ments were entitled to the most careful consideration, and this 
again necessitated research in the earlier history of Greek astronomy 
with the view of tracing every step in the progress towards the 
true Copernican theory. The first to substitute another centre for 
the earth in the celestial system were the Pythagoreans, who made 
the earth, like the sun, moon, and planets, revolve round the central 
fire ; and, when once my study of the subject had been carried 
back so far, it seemed to me that the most fitting introduction to 
Aristarchus would be a sketch of the whole history of Greek 
astronomy up to his time. As regards the newest claim made by 
Schiaparelli on behalf of Heraclides of Pontus, I hope I have 
shown that the case is not made out, and that there is still no 
reason to doubt the unanimous testimony of antiquity that 
Aristarchus was the real originator of the Copernican hypothesis. 
In the century following Copernicus no doubt was felt as to 



PREFACE ν 

identifying Aristarchus with the latter hypothesis. Libert Fro- 
mond, Professor of Theology at the University of Louvain, who 
tried to refute it, called his work Anti- Aristarchus (Antwerp, 
163 1). In 1644 Roberval took up the cudgels for Copernicus in a 
book the full title of which is Aristarchi Samii de mundi systemate 
partibus et motibus eiusdem libellus. Adiectae sunt JE* P. de 
Roberval) Mathem. Scient. in Collegio Regio Franciae Professoris, 
notae in eundem libellurn. It does not appear that experts were 
ever deceived by this title, although Baillet (Jugemens des Savans) 
complained of such disguises and would have had Roberval call his 
work Aristarchus G alius, 'the French Aristarchus,' after the 
manner of Vieta's Apollonius Gallus and Snellius's Eratosthenes 
Batavus. But there was every excuse for Roberval. The times 
were dangerous. Only eleven years before seven Cardinals had 
forced Galilei to abjure his ■ errors and heresies ' ; what wonder 
then that Roberval should take the precaution of publishing his 
views under another name? 

Voltaire, as is well known, went sadly wrong over Aristarchus 
(Dictionnaire Ρ hilosophique , s. v. ' Systeme '). He said that Ari- 
starchus ' is so obscure that Wallis was obliged to annotate him 
from one end to the other, in the effort to make him intelligible ', 
and further that it was very doubtful whether the book attributed 
to Aristarchus was really by him. Voltaire (misled, it is true, by 
a wrong reading in a passage of Plutarch, De facie in orbe lunae, 
c. 6) goes on to question whether Aristarchus had ever propounded 
the heliocentric hypothesis ; and it is clear that the treatise which 
he regarded as suspect was Roberval's book, and that he confused 
this with the genuine work edited by Wallis. Nor could he have 
looked at the latter treatise in any but a very superficial way, or 
he would have seen that it is not in the least obscure, and that the 
commentary of Wallis is no more elaborate than would ordinarily 
be expected of an editor bringing out for the first time, with the 
aid of MSS. not of the best, a Greek text and translation of a 
mathematical treatise in which a number of geometrical propositions 
are assumed without proof and therefore require some elucidation. 



vi PREFACE 

There is no doubt whatever of the genuineness of the work. 
Pappus makes substantial extracts from the beginning of it and 
quotes the main results. Apart from its astronomical content, it is 
of the greatest interest for its geometry. Thoroughly classical in 
form and language, as befits the period between Euclid and 
Archimedes, it is the first extant specimen of pure geometry used 
with a trigonometrical object, and in this respect is a sort of fore- 
runner of Archimedes' Measurement of a Circle. I need therefore 
make no apology for offering to the public a new Greek text with 
translation and the necessary notes. 

In conclusion I desire to express my best acknowledgements to 
the authorities of the Vatican Library for their kindness in allowing 
me to have a photograph of the best MS. of Aristarchus which 
forms part of the magnificent Codex Vaticanus Graecus 204 of the 
tenth century, and to Father Hagen of the Vatican Observatory 
for his assistance in the matter. 

T. L. H. 



CONTENTS 

PART I 
GREEK ASTRONOMY TO ARISTARCHUS OF SAMOS 

CHAPTER PAGES 

I. Sources of the History 1-6 

* II. Homer and Hesiod 7-1 1 

III. Thales 12-23 

IV. Anaximander 24-39 

V. Anaximenes 40-45 

,, VI. Pythagoras 46-ςι 

VII. Xenophanes 52-58 

VIII. Heraclitus 59-61 

IX. Parmenides . 62-77 

f X. Anaxagoras 78-85 

' XI. Empedocles 86-93 

XII. The Pythagoreans ...... 94-120 

XIII. The Atomists, Leucippus and Democritus . 121-129 

# XIV. Oenopides 130-133 

XV. Plato 134-189 

XVI. The Theory of Concentric Spheres — Eudoxus, 

Callippus, and Aristotle 190-224 

XVII. Aristotle {continued) 225-248 

XVIII. Heraclides of Pontus 249-283 

_ XIX. Greek Months, Years, and Cycles . . . 284-297 

PART II 

ARISTARCHUS ON THE SIZES AND DISTANCES 
OF THE SUN AND MOON 

I. ARISTARCHUS OF SAMOS 299-3 1 6 

II. The Treatise on Sizes and Distances — History 

of the Text and Editions .... 317-327 

III. Content of the Treatise ..... 328-336 

IV. Later Improvements on Aristarchus's Calcula- 

tions 337-350 

Greek Text, Translation, and Notes .... 351-414 

Index ........... 415-425 



CORRIGENDUM 

P. 179, lines 26 and 31. It appears that προχωρήσεις, not προσχωρήσεις, is 
the correct reading in Timaeus 40 c. The meaning of προχωρήσεις is of course 
' forward movements ', but the change to this reading does not make it any 
the more necessary to take επανακνκλήσεις in the sense of retrogradations ; on 
the contrary, a 'forward movement' and a 'returning of the circle upon itself 
are quite natural expressions for the different stages of one simple circular 
motion. Cf. also Republic 617 B, where επανακνκλούμενον is used of the 
1 counter-revolution 3 of the planet Mars ; what is meant is a simple circular 
revolution in a sense contrary to that of the fixed stars, and there is no suggestion 
of retrogradations. 



PART I 

GREEK ASTRONOMY TO ARISTARCHUS OF SAMOS 



SOURCES OF THE HISTORY 

The history of Greek astronomy in its beginnings is part of the 
history of Greek philosophy, for it was the first philosophers, 
Ionian, Eleatic, Pythagorean, who were the first astronomers. 
Now only very few of the works of the great original thinkers 
of Greece have survived. We possess the whole of Plato and, say, 
half of Aristotle, namely, those of his writings which were intended 
for the use of his school, but not those which, mainly composed 
in the form of dialogues, were in a more popular style. But the 
whole of the pre-Socratic philosophy is one single expanse of 
ruins; 1 so is the Socratic philosophy itself, except for what we 
can learn of it from Plato and Xenophon. 

But accounts of the life and doctrine of philosophers begin to 
appear quite early in ancient Greek literature (cf. Xenophon, who 
was born between 430 and 425 B. c.) ; and very valuable are the 
allusions in Plato and Aristotle to the doctrines of earlier philo- 
sophers ; those in Plato are not very numerous, but he had the 
power of entering into the thoughts of other men and, in stating 
the views of early philosophers, he does not, as a rule, read into 
their words meanings which they do not convey. Aristotle, on the 
other hand, while making historical surveys of the doctrines of his 
predecessors a regular preliminary to the statement of his own, 
discusses them too much from the point of view of his own system ; 
often even misrepresenting them for the purpose of making a contro- 
versial point or finding support for some particular thesis. 

From Aristotle's time a whole literature on the subject of the 
older philosophy sprang up, partly critical, partly historical. This 

1 Gomperz, Griechische Denker, i 3 , p. 419. 
1410 Β 



2 SOURCES OF THE HISTORY parti 

again has perished except for a large number of fragments. Most 
important for our purpose are the notices in the Doxographi Graeci, 
collected and edited by Diels. 1 The main source from which these 
retailers of the opinions of philosophers drew, directly or indirectly, 
was the great work of Theophrastus, the successor of Aristotle, 
entitled Physical Opinions (Φυσικών δόξων Τη). It would appear 
that it was Theophrastus's plan to trace the progress of physics 
from Thales to Plato in separate chapters dealing severally with 
the leading topics. First the leading views were set forth on broad 
lines, in groups, according to the affinity of the doctrine, after 
which the differences between individual philosophers within the 
same group were carefully noted. In the First Book, however, 
dealing with the Principles, Theophrastus adopted the order of the 
various schools, Ionians, Eleatics, Atomists, &c, down to Plato, 
although he did not hesitate to connect Diogenes of Apollonia and 
Archelaus with the earlier physicists, out of their chronological 
order ; chronological order was indeed, throughout, less regarded 
than the connexion and due arrangement of subjects. This work 
of Theophrastus was naturally the chief hunting-ground for those 
who collected the ' opinions ' of philosophers. There was, however, 
another main stream of tradition besides the doxographic ; this 
was in the different form of biographies of the philosophers. The 
first to write a book of * successions ' (διαδοχαί) of the philosophers 
was Sotion (towards the end of the third century B. C.) ; others 
who wrote 'successions' were a certain Antisthenes (probably 
Antisthenes of Rhodes, second century B.C.), Sosicrates, and 
Alexander Polyhistor. These works gave little in the way ot 
doxography, but were made readable by the incorporation of 
anecdotes and apophthegms, mostly unauthentic. The work 
of Sotion and the 'Lives of Famous Men' by Satyrus (about 
1 60 B.C.) were epitomized by Heraclides Lembus. Another writer 
of biographies was the Peripatetic Hermippus of Smyrna, known as 
the Callimachean, who wrote about Pythagoras in at least two Books, 
and is quoted by Josephus as a careful student of all history. 2 Our 
chief storehouse of biographical details derived from these and all 
other available sources is the great compilation which goes by the 

1 Doxographi Graeci, ed. Diels, Berlin, G. Reimer, 1879. 

2 Doxographi Graeci (henceforth generally quoted as D. G.), p. 1 5 1. 



CH. I 



SOURCES OF THE HISTORY 



name of Diogenes Laertius (more properly Laertius Diogenes). It 
is a compilation made in the most haphazard way, without the 
exercise of any historical sense or critical faculty. But its value 
for us is enormous because the compiler had access to the whole 
collection of biographies which accumulated from Sotion's time to 
the first third of the third century A. D. (when Diogenes wrote), and 
consequently we have in him the whole residuum of this literature 
which reached such dimensions in the period. 

In order to show at a glance the conclusions of Diels as to the 
relation of the various representatives of the doxographic and 
biographic traditions to one another and to the original sources 
I append a genealogical table * : 

The 



Simplicius 




^ SoHon 
^ια&οχη των φιλοσόφων) 



VefustePI&o'KSO&jQ Stoic Epitome 

from Iheophndus 



Epitome of 
Aerius(iooA.D) 



Theodpretos^-so^.R; 
φνληνικων παθημάτων 
eepcwttuTuo^ 5k<IV) 

NemejiusfcooA.D) 
(de nalwa hommii) 




Philodemus 
(De pietale) 



Hippolytas, Diogenes Laerhas 



Cicero 



Clemens Alexandrians. 



Stotaeus (fclogae 
physicae) &bour47o AD. 



Placita. PhiLosophorum 
(7 ad Cent. Α.β) 



Meletius 



(Pscwdo-PlukreK) 



Philo 
(Do. providenliq}/ 



Atoea&jordS Q77 AD.) 
(supplic pro Cnrislianis) 




(Hisloria philosopha) 
(_5ooA.D) 



Pseudo-Justinus 
C2ndCenhA.R) 

(cbhortalw ad gentiles) 

Cyrilius 0th. cear. A. Ό.) 
(contra Julianum.) 



EusebitiS 
(4rh..Ceat.AD) 
(fuanqclica prcxeparaiio 
Bks.XIV.XV) 



Fig. ι 



1 Cf. Gunther in Windelband, Gesch. der alten Philosophic (I wan von Miiller's 
Handbuch der klassischen Altertumswissenschaft ! , Band v. i), 1894, p. 275. 



Β 2 



4 SOURCES OF THE HISTORY parti 

Only a few remarks need be added. 'Vetusta Placita' is the 
name given by Diels to a collection which has disappeared, but 
may be inferred to have existed. It adhered very closely to 
Theophrastus, though it was not quite free from admixture of 
other elements. It was probably divided into the following main 
sections: I. De principiis ; II. De mundo; III. De sublimibus ; 
IV. De terrestribus ; V. De anima ; VI. De corpore. The date 
is inferred from the facts that the latest philosophers mentioned 
in it were Posidonius and Asclepiades, and that Varro used it. 
The existence of the collection of Aetius (De placitis, nepl 
άρ^σκοντων) is attested by Theodoretus (Bishop of Cyrus), who 
mentions it as accessible, and who certainly used it, since his 
extracts are more complete and trustworthy than those of the 
Placita Philosophorum and Stobaeus. The compiler of the Placita 
was not Plutarch, but an insignificant writer of about the middle 
of the second century A. D., who palmed them off as Plutarch. 
Diels prints the Placita in parallel columns with the corresponding 
parts of the Eclogae, under the title of Aetii Placita ; quotations 
from the other writers who give extracts are added in notes 
at the foot of the page. So far as Cicero deals with the earliest 
Greek philosophy, he must be classed with the doxographers ; both 
he and Philodemus (De pietate, wepl earejSetay, fragments of which 
were discovered on a roll at Herculaneum) seem alike to have used a 
common source which went back to a Stoic epitome of Theophrastus, 
now lost. 

The greater part of the fragment of the Pseudo-Plutarchian 
στρωματεΐς given by Eusebius in Book I. 8 of the Praeparatio 
Evangelica comes from an epitome of Theophrastus, arranged 
according to philosophers. The author of the Stromateis, who 
probably belonged to the same period as the author of the Placita, 
that is, about the middle of the second century A. D., confined 
himself mostly to the sections de principle•, de mundo, de astris ; 
hence some things are here better preserved than elsewhere ; cf. 
especially the notice about Anaximander. 

The most important of the biographical doxographies is that of 
Hippolytus in Book I of the Refutation of all Heresies (the sub- 
title of the particular Book is φιλοσόφου μένα), probably written 
between 223 and 235 A.D. It is derived from two sources. The 



ch.i SOURCES OF THE HISTORY 5 

one was a biographical compendium of the διαδοχή type, shorter 
and even more untrustworthy than Diogenes Laertius, but con- 
taining excerpts from Aristoxenus, Sotion, Heraclides Lembus, 
and Apollodorus. The other was an epitome of Theophrastus. 
Hippolytus's plan was to take the philosophers in order and then 
to pick out from the successive sections of the epitome of Theo- 
phrastus the views of each philosopher on each topic, and insert 
them in their order under the particular philosopher. So carefully 
was this done that the divisions of the work of Theophrastus can 
practically be restored. 1 Hippolytus began with the idea of dealing 
with the chief philosophers only, as Thales, Pythagoras, Empedocles, 
Heraclitus. For these he had available only the inferior (biographical) 
source. The second source, the epitome of Theophrastus, then 
came into his hands, and, beginning with Anaximander, he proceeded 
to make a most precious collection of opinions. 

Another of our authorities is Achilles (not Tatius), who wrote 
an Introduction to the Phaenomena of A rat us. 2 Achilles' date is 
uncertain, but he probably lived not earlier than the end of the 
second century A. D., and not much later. The foundation of 
Achilles' commentary was a Stoic compendium of astronomy, 
probably by Eudorus, which in its turn was extracted from a work 
by Diodorus of Alexandria, a pupil of Posidonius. But Achilles 
drew from other sources as well, including the Pseudo-Plutarchian 
Placita\ he did not hesitate to alter his extracts from the latter, 
and to mix alien matter with them. 

The opinions noted by the Doxographi are largely incorporated 
in Diels' later work Die Fragmente der Vorsokratiker? 

For the earlier period from Thales to Empedocles, Tannery gives 
a translation of the doxographic data and the fragments in his 
work Pour Vhistoire de la science Hellene, de Thales a Empedocle, 
Paris, 1887 ; taking account as it does of all the material, this work is 
the best and most suggestive of the modern studies of the astronomy 
of the period. Equally based on the Doxographi, Max Sartorius's 
dissertation Die Entwicklung der Astronomie bei den Griechen bis 

1 Diels, Doxographi Graeci, p. 153. 

2 Excerpts from this are preserved in Cod. Laurentian. xxviii. 44, and are 
included in the Uranologium of Petavius, 1630, pp. 121-64, &c. 

8 Second edition in two vols, (the second in two parts), Berlin, 1906-10. 



6 SOURCES OF THE HISTORY 

Anaxagoras und Empedokles (Halle, 1883) is a very concise and 
useful account. Naturally all or nearly all the material is also to 
be found in the monumental work of Zeller and in Professor Burnet's 
Early Greek Philosophy (second edition, 1908); and picturesque, 
if sometimes too highly coloured, references to the astronomy of 
the ancient philosophers are a feature of vol. i of Gomperz's 
Griechische Denker (third edition, 191 1). 

Eudemus of Rhodes (about 330 Β. a), a pupil of Aristotle, wrote 
a History of Astronomy (as he did a History of Geometry), which 
is lost, but was the source of a number of notices in other writers. 
In particular, the very valuable account of Eudoxus's and Callip- 
pus's systems of concentric spheres which Simplicius gives in his 
Commentary on Aristotle's De caelo is taken from Eudemus through 
Sosigenes as intermediary. A few notices from Eudemus's work 
are also found in the astronomical portion of Theon of Smyrna's 
Expositio rerum mat hematic arum ad legendum Platonem utilium^ 
which also draws on two other sources, Dercyllides and Adrastus. 
The former was a Platonist with Pythagorean leanings, who wrote 
a book on Plato's philosophy. His date was earlier than the time 
of Tiberius, perhaps earlier than Varro's. Adrastus, a Peripatetic 
of about the middle of the second century A.D., wrote historical 
and lexicographical essays on Aristotle ; he also wrote a commentary 
on the Timaeus of Plato, which is quoted by Proclus as well as by 
Theon of Smyrna. 

1 Edited by E. Hiller (Teubner, 1878). 



II 

HOMER AND HESIOD 

We take as our starting-point the conceptions of the structure 
of the world which are to be found in the earliest literary monuments 
of Greece, that is to say, the Homeric poems and the works of 
Hesiod. In their fundamental conceptions Homer and Hesiod 
agree. The earth is a flat circular disc; this is not stated in so 
many words, but only on this assumption could Poseidon from 
the mountains of Solym in Pisidia see Odysseus at Scheria on the 
further side of Greece, or Helios at his rising and setting descry 
his cattle on the island of Thrinakia. Round this flat disc, on the 
horizon, runs the river Oceanus, encircling the earth and flowing 
back into itself (άψόρροος); from this all other waters take their 
rise, that is, the waters of Oceanus pass through subterranean 
channels and appear as the springs and sources of other rivers. 
Over the flat earth is the vault of heaven, like a sort of hemi- 
spherical dome exactly covering it ; hence it is that the Aethiopians 
dwelling in the extreme east and west are burnt black by the sun. 
Below the earth is Tartarus, covered by the earth and forming 
a sort of vault symmetrical with the heaven ; Hades is supposed 
to be beneath the surface of the earth, as far from the height of 
the heaven above as from the depth of Tartarus below, Le. pre- 
sumably in the hollow of the earth's disc. The dimensions of the 
heaven and earth are only indirectly indicated ; Hephaestus cast 
down from Olympus falls for a whole day till sundown ; on the 
other hand, according to Hesiod, an iron anvil would take nine 
days to pass from the heaven to the earth, and again nine days 
from the earth to Tartarus. The vault of heaven remains for ever 
in one position, unmoved ; the sun, moon, and stars move round 
under it, rising from Oceanus in the east and plunging into it again 
in the west. We are not told what happens to the heavenly bodies 



8 HOMER AND HESIOD parti 

between their setting and rising; they cannot pass round under 
the earth because Tartarus is never lit up by the sun ; possibly 
they are supposed to float round Oceanus, past the north, to the 
points where they next rise in the east, but it is only later writers 
who represent Helios as sleeping and being carried round on the 
water on a golden bed or in a golden bowl. 1 

Coming now to the indications of actual knowledge of astronomical 
facts to be found in the poems, we observe in Hesiod a considerable 
advance as compared with Homer. Homer mentions, in addition 
to the sun and moon, the Morning Star, the Evening Star, the 
Pleiades, the Hyades, Orion, the Great Bear (' which is also called 
by the name of the Wain, and which turns round on the same spot 
and watches Orion ; it alone is without lot in Oceanus's bath ' 2 ), 

1 Athenaeus, Deipnosoph. xi. 38-9. 

2 It seems that some of the seven principal stars of the Great Bear do now 
set in the Mediterranean, e.g., in places further south in latitude than Rhodes 
(lat. 36 ), γ, the hind foot, as well as 77, the tip of the tail, and at Alexandria all 
the seven stars except a, the head. But this was not so in Homer's time. In 
proof of this, Sir George Greenhill (in a lecture delivered in 1910 to the Hellenic 
Travellers' Club) refers to calculations made by Dr. J. B. Pearson of the effect 
of Precession in the interval since 750 B.C., a date taken 'without prejudice' 
{Proceedings of the Cambridge Philosophical Society, 1877 and 1881), and to the 
results obtained in a paper by J. Gallenmuller, Der Fixsternhimmel jetzt undin 
Homers Zeiten mit zwei Sternkarten (Regensburg, 1884/85). Gallenmiiller's 
charts are for the years 900 B.C. and A.D. 1855 respectively, and the chart for 
900 B.C. shows that the N.P. D. of both β, the fore-foot, and 77, the tip of the 
tail, was then about 25 . But we also find convincing evidence in the original 
writings of the Greek astronomers. Hipparchus {In Arati et Eudoxi phaeno- 
mena commentariorum libri Ires, ed. Manitius, 1894, p. 114. 9-10) observes that 
Eudoxus [say, in 380 B.C., or 520 years later than the date to which Gallen- 
miiller's chart refers] made the fore-foot (β) about 24 , and the hind-foot (y) 
about 25 , distant from the north pole. This was perhaps not very accurate ; 
for Hipparchus says (ibid., p. 30. 2-8), ' As regards the north pole, Eudoxus is in 
error in stating that " there is a certain star which always remains in the same 
spot, and this star is the pole of the universe " ; for in reality there is no star at 
all at the pole, but there is an empty space there, with, however, three stars 
near to it [probably α and κ of Draco and β of the Little Bear], and the point at 
the pole makes with these three stars a figure which is very nearly square, as 
Pytheas of Massalia stated.' (Pytheas, the great explorer of the northern seas, 
was a contemporary of Aristotle, and perhaps some forty years later than 
Eudoxus.) But, as Hipparchus himself (writing in this case not later than 
1 34 B. c.) makes the angular radius of the ' always-visible circle ' 2>7° at Athens 
and 36 at Rhodes (ibid., pp. 112. 16 and 114. 24-6), it is evident that in 
Eudoxus's time the whole of the Great Bear remained well above the horizon. 
A passage of Proclus {Hypotyfiosis, c. 7, §§ 45-8, p. 234, ed. Manitius) is not 
without interest in this connexion. He is trying to controvert the theory of 
astronomers that the fixed stars themselves have a movement about the pole 
of the ecliptic (as distinct from the pole of the universe) of about i° in 100 years 



ch.ii HOMER AND HESIOD 9 

Sirius ('the star which rises in late summer . . , which is called 
among men " Orion's dog " ; bright it shines forth, yet is a baleful 
sign, for it brings to suffering mortals much fiery heat '), the ' late- 
setting Bootes ' (the ' ploughman ' driving the Wain, i. e. Arcturus, 
as Hesiod was the first to call it). Since the Great Bear is said 
to be the only constellation which never sets, we may perhaps 
assume that the stars and constellations above named are all that 
were definitely recognized at the time, or at least that the Bear 
was the only constellation recognized in the northern sky. There 
is little more that can be called astronomy in Homer. There are 
vague uses of astronomical phenomena for the purpose of fixing 
localities or marking times of day or night ; as regards the day, 
the morning twilight, the rising and setting of the sun, midday, 
and the onset of night are distinguished ; the night is divided into 
three thirds. Aristotle was inclined to explain Helios's seven herds 
of cattle and sheep respectively containing 50 head in each herd 
(i. e. 350 in all of each sort) as a rough representation of the number 
of days in a year. Calypso directed Odysseus to sail in such a way 
as to keep the Great Bear always on his left. One passage, 1 
relating to the island called Syrie, c which is above Ortygia where 
are the turnings (τροπαί) of the sun \ is supposed by some to refer 
to the solstices, but there is no confirmation of this by any other 
passage, and it seems safer to take 'turning' to mean the turn 
which the sun takes at setting, when of course he begins his return 
journey (travelling round Oceanus or otherwise) to the place of his 

(this is Ptolemy's estimate). ' How is it', says Proclus, 'that the Bears, which 
have always been visible above the horizon through countless ages, still remain 
so, if they move by one degree in 100 years about the pole of the zodiac, which 
is different from the world-pole ; for, if they had moved so many degrees as this 
would imply, they should now no longer graze (παραζίειν) the horizon but should 
partly set ' ! This passage, written (say) 840 years after Eudoxus's location of β and 
γ of the Great Bear, shows that the Great Bear was then much nearer to setting 
than it was in Eudoxus's time, and the fact should have made Proclus speak with 
greater caution. [The star which Eudoxus took as marking the north pole has 
commonly been supposed to be β of the Little Bear ; but Manitius {Hipparchi in 
Arati et Eicdoxi phaen. comment., 1894, p. 306), as the result of studying a 
* Precession -globe ' designed by Prof. Haas of Vienna, considers that it was 
certainly a different star, namely, ' Draconis 16,' which occupies a position 
determined as the intersection of (1) a perpendicular from our Polar Star to the 
straight line joining κ and λ of Draco and (2) the line joining y and β of the 
Little Bear and produced beyond β.] 
1 Odyssey xv. 403-4. 



ίο HOMER AND HESIOD parti 

rising, in which case the island would simply be situated on the 
western horizon where the sun sets. 1 

Hesiod mentions practically the same stars as Homer, the 
Pleiades, the Hyades, Orion, Sirius, and Arcturus. But, as might 
be expected, he makes much more use than Homer does of celestial 
phenomena for the purpose of determining times and seasons in the 
year. Thus, e. g., he marked the time for sowing at the beginning 
of winter by the setting of the Pleiades in the early twilight, or 
again by the early setting of the Hyades or Orion, which means 
the 3rd, 7th, or 15th November in the Julian calendar according to 
the particular stars taken; 2 the time for harvest he fixed by the 
early rising of the Pleiades, which means the Julian 19th of May ; 3 
threshing-time he marked by the early rising of Orion (Julian 9th 
of July), vintage-time by the early rising of Arcturus (Julian 18th 
of September), and so on. 4 With Hesiod, Spring begins with the 
late rising of Arcturus ; this would in his time and climate be the 
24th February of the Julian calendar, or 57 days after the winter 
solstice, which in his time would be the 29th December. He him- 
self makes Spring begin 60 days after the winter solstice ; he may 
be intentionally stating a round figure, but, if he made an error of 

1 Martin has discussed the question at considerable length (' Comment 
Homere s'orientait ' in Me'?noires de PAcademie des Inscriptions et Belles- 
Lettres, xxix, Pt. 2, 1879, pp. 1-28). He strongly holds that τροπαι ψλίοιο can 
only mean the solstice, that by this we must also understand the summer 
solstice, and that the expression 6Θ1 τροπαΐ ψλίοιο must therefore be in the 
direction of the place on the horizon where the sun sets at the summer solstice, 
i.e. west-north-west. Martin's ground is his firm conviction that τροπαι ψλίοιο 
has never, in any Greek poet or prose writer, any other than the technical 
meaning of ' solstice \ This is, however, an assumption not susceptible of proof; 
and Martin is not very successful in his search for confirmation of his view. 
Identifying Ortygia with Delos, and Syrie with Syra or Syros, he admits that 
the southern part of Syra is due west of the southern part of Delos ; only the 
northern portion of Syra stretches further north than the northern portion of 
Delos ; therefore, geographically, either west or west-north-west would describe 
the direction of Syra relatively to Ortygia well enough. Of the Greek com- 
mentators, Aristarchus of Samothrace and Herodian of Alexandria take τροπαί 
to mean ' setting ' simply ; Martin is driven therefore to make the most he can 
of Hesychius who (s.v. Όρτνγίη) gives as an explanation τούτο 8e έστιν οπού ai 
δυσβίί άρχονται, ' This is where the settings commence ', which Martin interprets 
as meaning 'where the sun sets at the commencement of the Greek year', which 
was about the time of the summer solstice ; but this is a great deal to get out 
of ' commencement of setting \ 

2 Ideler, Handbuch der ?nathe?natischen und technischen Chronologie, 1825, 
i, pp. 242, 246. 

a Ibid. p. 242. 4 Ibid. pp. 246, 247. 



CH. π HOMER AND HESIOD n 

three days, it would not be surprising, seeing that in his time there 
were no available means for accurately observing the times of the 
solstices. His early summer (θέρος), as distinct from late summer 
(όχτώρα), he makes, in like manner, end 50 days after the sum- 
mer solstice. Thus he was acquainted with the solstices, but he 
says nothing about the equinoxes, and only remarks in one place 
that in late summer the days become shorter and the nights longer. 
From the last part of the Works and Days we see that Hesiod had 
an approximate notion of the moon's period ; he puts it at 30 days, 
and divides the month into three periods of ten days each. 1 

Hesiod was also credited with having written a poem under the 
title of ' Astronomy '. A few fragments of such a poem are pre- 
served; 2 Athenaeus, however, doubted whether it was Hesiod's 
work, for he quotes ' the author of the poem rt Astronomy " which 
is attributed to Hesiod ' as always speaking of Pleiades. Pliny 
observes that ' Hesiod (for an Astrology is also handed down 
under his name) stated that the matutinal setting of the Vergiliae 
[Pleiades] took place at the autumnal equinox, whereas Thales 
made the time 25 days from the equinox \ 3 The poem was thought 
to be Alexandrine, but has recently been shown to be old ; perhaps, 
if we may judge by the passage of Pliny, it may be anterior to 
Thales. 

1 Sartorius, op. cit., p. 16 ; Ideler, i, p. 257. 

2 Diels, Vorsokratiker, ii 2 . 1, 1907, pp. 499, 500. 

3 Pliny, N. H. xviii, c. 25, § 213 ; Diels, loc. cit. 



Ill 
THALES 

Such astronomy as we find in Homer and Hesiod was of the 
merely practical kind, which uses the celestial recurrences for the 
regulation of daily life ; but, as the author of the Epinomis says, 
' the true astronomer will not be the man who cultivates astronomy 
in the manner of Hesiod and any other writers of that type, concern- 
ing himself only with such things as settings and risings, but the 
man who will investigate the seven revolutions included in the eight 
revolutions and each describing the same circular orbit [i. e. the 
separate motions of the sun, moon, and the five planets combined 
with the eighth motion, that of the sphere of the fixed stars, or the 
daily rotation], which speculations can never be easily mastered by 
the ordinary person but demand extraordinary powers \ The history 
of Greek astronomy in the sense of astronomy proper, the astronomy 
which seeks to explain the heavenly phenomena and their causes, 
begins with Thales. 

Thales of Miletus lived probably from about 624 to 547 B. C. 
(though according to Apollodorus he was born in 640/39). Accord- 
ing to Herodotus, his ancestry was Phoenician ; his mother was 
Greek, to judge by her name Cleobuline, while his father's name, 
Examyes, is Carian, so that he was of mixed descent. In 582/1 B. c. 
he was declared one of the Seven Wise Men, and indeed his ver- 
satility was extraordinary ; statesman, engineer, mathematician and 
astronomer, he was an acute business man in addition, if we may 
believe the story that, wishing to show that it was easy to get rich, 
he took the opportunity of a year in which he foresaw that there 
would be a great crop of olives to get control of all the oil-presses 
in Miletus and Chios in advance, paying a low rental when there 
was no one to bid against him, and then, when the accommodation 
was urgently wanted, charging as much as he liked for it, with the 
result that he made a large profit. 1 For his many-sided culture he 

1 Aristotle, Politics i. II. 9, 1259 a 6-17. 



THALES 



τ 3 



was indebted in great measure to what he learnt on long journeys 
which he took, to Egypt in particular ; it was in Egypt that he saw 
in operation the elementary methods of solving problems in prac- 
tical geometry which inspired him with the idea of making 
geometry a deductive science depending on general propositions ; 
and he doubtless assimilated much of the astronomical knowledge 
which had been accumulated there as the result of observations 
recorded through long centuries. -""Τ 

Thales* claim to a place in the history of scientific astronomy ~λ/ λ ' 
depends almost entirely on one achievement attributed to him, that 
of predicting an eclipse of the sun. | There is no trustworthy / 
evidence of any other discoveries, or even of any observations, 
made by him, although one would like to believe the story, quoted 
by Plato, 1 that, when he was star-gazing and fell into a well in con- 
sequence, he was rallied ' by a clever and pretty maid-servant from 
Thrace' 2 for being so ' eager to know what goes on in the heavens 
when he could not see what was in front of him, nay, at his very feet '. 

But did Thales predict a solar eclipse? The story is entirely 
rejected by Martin. 3 He points out that, while the references to 
the prediction do not exactly agree, it is in fact necessary, if the 
occurrence of a solar eclipse at any specified place on the earth's 
surface is to be predicted with any prospect of success, to know ~-y\ 
more of the elements of astronomy than Thales could have known, 
and in particular to allow for parallax, which was not done until 
much later, and then only approximately, by Hipparchus. Further, 
if the prophecy had rested on any scientific basis, it is incredible 
that the basis should not have been known and been used by later 
Ionian philosophers for making other similar predictions, whereas 
we hear of none such in Greece for two hundred years. Indeed, 
only one other supposed prediction of the same kind is referred to. 
Plutarch 4 relates that, when Plato was on a visit to Sicily and stay- 
ing with Dionysius, Helicon of Cyzicus, a friend of Plato's, foretold 
a solar eclipse (apparently that which took place on 12th May, 

1 Theaetetus 174 A; cf. Hippolytus, Refut. i. 1. 4 {D. G. p. 555. 9-12). 

2 There is another version not so attractive, according to which [Diog. Laert. 
i. 34], being taken out of the house by an old woman to look at the stars, he fell 
into a hole and was reproached by her in similar terms. This version might 
suggest that it was the old woman who was the astronomer rather than Thales. 

3 Revue Arche'ologique, ix, 1864, pp. 181 sq. 4 Life of Dion, c. 19, p. 966 A. 






iA 



la THALES part ι 

361 B.C.), 1 and, when this took place as predicted, the tyrant was 
filled with admiration and made Helicon a present of a talent of 
silver. This story is, however, not confirmed by any other evidence, 
and the necessary calculations would have been scarcely less im- 
J possible for Helicon than for Thales. Martin's view is that both 
Thales and Helicon merely explained the cause of solar eclipses 
and asserted the necessity of their recurrence within certain limits 
of time, and that these explanations were turned by tradition into 
predictions. In regard to Thales, Martin relies largely on the word- 
ing of a passage in Theon of Smyrna, where he purports to quote 
Eudemus ; c Eudemus ', he says, c relates in his Astronomies that . . . 
Thales was the first to discover (eupe πρώτος understood) the 
eclipse of the sun and the fact that the sun's period with respect 
to the solstices is not always the same \ 2 and the natural mean- 
ing of the first part of the sentence is that Thales discovered^ 
the explanation and the cause of a solar eclipse. It is true that 
Diogenes Laertius says that { Thales appears, according to some, to 
have been the first to study astronomy and to predict both solar 
eclipses and solstices, as Eudemus says in his History of Astronomy ', 3 
and Diogenes must be quoting from the same passage as Theon ; 
but it is pretty clear, as Martin says, that he copied it inaccurately 
and himself inserted the word (npoeLnetv) referring to predictions ; 
indeed the word ' predict ' does not go well with * solstices ', and is 
suspect for this reason. Nor does any one credit Thales with having 
predicted more than one eclipse. No doubt the original passage 
spoke of ' eclipses ' and ' solstices ' in the plural and used some word 
like 'discover' (Theon's word), not the word 'predict'. And I 
think Martin may reasonably argue from the passage of Diogenes 
that the words ' according to some ' are Eudemus's words, not his 
own, and therefore may be held to show that the truth of the 
tradition was not beyond doubt. 4 

1 Boll, art. ' Finsternisse ' in Pauly-Wissowa's Real-Encyclofiadie der 
classischen Altertumswissenschaft, vi. 2, 1909, pp. 2356-7 ; Ginzel, Handbuch 
der mathematischen und technischen Chronologie, vol. ii, 191 1, p. 527. 

2 Theon of Smyrna, ed. Hiller, p. 198. 14-18. 
8 Diog. L. i.23 {Vorsokratiker> i 2 , p. 3. 19-21). 
4 There is, however, yet another account purporting to be based on Eudemus. 

Clement of Alexandria (Stromat. i. 65) says : 'Eudemus observes in his History 
of Astronomy that Thales predicted the eclipse of the sun which took place at 
the time when the Medes and the Lydians engaged in battle, the king of the 



ch.iii THALES 15 

Nevertheless, as Tannery observes, 1 Martin's argument can 
hardly satisfy us so far as it relates to Thales. The evidence 
that Thales actually predicted a solar eclipse is as conclusive as 
we could expect for an event belonging to such remote times, for 
Diogenes Laertius quotes Xenophanes as well as Herodotus as 
having admired Thales' achievement, and Xenophanes was almost 
contemporary with Thales. We must therefore accept the fact as 
historic, and it remains to inquire In what sense or form, and on 
what ground, he made his prediction. The accounts of it vary. 
Herodotus says 2 that the Lydians and the Medes continued their 
war, and ' when, in the sixth year, they encountered one another, it 
fell out that, after they had joined battle, the day suddenly turned ^ 

into night. Now that this transformation of day (into night) would 
occur was foretold to the Ionians by Thales of Miletus, who fixed 
as the limit of time this very year in which the change actually 
took place.' 3 The prediction was therefore at best a rough one, 

Medes being Cyaxares, the father of Astyages, and Alyattes, the son of Croesus, 
being the king of the Lydians ; and the time was about the 50th Olympiad [580- 
577]•' The last sentence was evidently taken from Tatian 41 ; but, if the rest of 
the passage correctly quotes Eudemus, it would appear that there must have 
been two passages in Eudemus dealing with the subject. 

1 Tannery, Pour thistoire de la science Hellene, p. 56. 

2 Herodotus, i. 74. 

3 Other references are as follows: Cicero, De Divinatione i. 49. 112, observes 
that Thales was said to have been the first to predict an eclipse of the sun, which 
eclipse took place in the reign of Astyages ; Pliny, N.H. ii, c. 12, § 53, 'Among 
the Greeks Thales first investigated (the cause of the eclipse) in the fourth year of 
the 48th Olympiad [585/4 B.C.], having predicted an eclipse of the sun which 
took place in the reign of Alyattes in the year 170 A.u.c.' ; Eusebius, Chron. 
(Hieron.), under year of Abraham 1433, 'An eclipse of the sun, the occurrence of 
which Thales had predicted : a battle between Alyattes and Astyages \ The 
eclipse so foretold is now most generally taken to be that which took place on 
the (Julian) 28th May, 585. A difficulty formerly felt in regard to this date 
seems now to have been removed. Herodotus (followed by Clement) says that 
the eclipse took place during a battle between Alyattes and Cyaxares. Now, 
on the usual assumption, based on Herodotus's chronological data, that Cyaxares 
reigned from about 635 to 595, the eclipse of 585 B.C. must have taken place 
during the reign of his son ; and perhaps it was the knowledge of this fact which 
made Eusebius say that the battle was between Alyattes and Astyages. But it 
appears that Herodotus's reckoning was affected by an error on his part in taking 
the fall of the Median kingdom to be coincident with Cyrus's accession to 
the throne of Persia, and that Cyaxares really reigned from 624 to 584, and 
Astyages from 584 to 550 B.C. (Ed. Meyer in Pauly-Wissowa's Real-Encyclo- 
fiadie, ii, 1896, p. 1865, &c.) ; hence the eclipse of 585 B.C. would after all come 
in Cyaxares' reign. Of two more solar eclipses which took place in the reign of 
Cyaxares one is ruled out, that of 597 B.C., because it took place at sunrise, which 
would not agree with Herodotus's story. The other was on 30th September, 610, 
and, as regards this, Bailly and Oltmanns showed that it was not total on the 







1 6 ΤΗ ALES parti 

since it only specified that the eclipse would occur within a certain 
year ; and the true explanation seems to be that it was a prediction 
of the same kind as had long been in vogue with the Chaldaeans. 
That they had a system enabling them to foretell pretty accurately 
the eclipses of the moon is clear from the fact that some of the 
eclipses said by Ptolemy 1 to have been observed in Babylon were so 
partial that they could hardly have been noticed if the observers had 
not been to some extent prepared for them. Three of the eclipses 
mentioned took place during eighteen months in the years 721 
and 720. It is probable that the Chaldaeans arrived at this method 
of approximately predicting the times at which lunar eclipses would 
occur by means of the period of 223 lunations, which was doubt- 
less discovered as the result of long-continued observations. This 
period is mentioned by Ptolemy 2 as having been discovered by 
astronomers ! still more ancient ' than those whom he calls ' the 
ancients'. 3 Now, while this method would serve well enough for 
lunar eclipses, it would very often fail for solar eclipses, because no 
account was taken of parallax. An excellent illustration of the 
way in which the system worked is on record ; it is taken from 
a translation of an Assyrian cuneiform inscription, the relevant words 
being the following : 

1. To the king my lord, thy servant Abil-istar. 

2. May there be peace to the king my lord. May Nebo and 
Merodach 

3. to the king my lord be favourable. Lerfgth of days, 

4. health of body and joy of heart may the great gods 

presumed field of battle (in Cappadocia), though it would be total in Armenia 
(Martin, Revue Archoologique, ix, 1864, pp. 183, 190). Tannery, however 
{Pour Vhistoire de la science hellene, p. 38), holds that the latter eclipse was that 
associated with Thales. The latest authorities (Boll, art. ' Finsternisse', in Pauly- 
Wissowa's Real-Encyclopadie, vi. 2, 1909, pp. 2353-4, and Ginzel, Spezieller Kanon 
der Sonnen- und Mondfinsternisse and Handbuch der mathematischen und tech- 
nischen Chronologie, vol. ii, 191 1, p. 525) adhere to the date 28th May, 585. 

1 Ptolemy, Syntaxis iv, c. 6 sq. 

2 Ptolemy, Syntaxis iv, c. 2, p. 270, 1 sq., ed. Heiberg. 

3 Suidas understands the Chaldaean name for this period to have been saros, 
but this seems to be a mistake. According to Syncellus {Chronograpkia, p. 17, 
A-B), Berosus expressed his periods in sars, ners, and sosses, a sar being 3,600 
years, while ner meant 600 years, and soss 60 years ; but we learn that the same 
words were also used to denote the same numbers of days respectively 
(Syncellus, p. 32 C). Nor were they used of years and days only ; in fact sar, 
tier, and soss were collective numerals simply, like our words ' gross ', ' score ', 
&c. (Cantor, Gesch. d. Mathematik, i 3 , p. 36). 

4 See George Smith, Assyrian Discoveries, p. 409. 



ch.iii ΤΗ ALES 17 

5. to the king my lord grant. Concerning the eclipse of the 

moon 

6. of which the king my lord sent to me; in the cities of 

Akkad, 

7. Borsippa, and Nipur, observations 

8. they made and then in the city of Akkad 

9. we saw part .... 
10. The observation was made and the eclipse took place. 

17. And when for the eclipse of the sun we made 

18. an observation, the observation was made and it did not take 

place. 

19. That which I saw with my eyes to the king my lord 

20. I send. This eclipse of the moon 

21. which did happen concerns the countries 

22. with their god all. Over Syria 

23. it closes, the country of Phoenicia, 

24. of the Hittites, of the people of Chaldaea, 

25. but to the king my lord it sends peace, and according to 

26. the observation, not the extending 

27. of misfortune to the king my lord 

28. may there be. 

It would seem, as Tannery says, 1 that these clever people knew 
how to turn their ignorance to account as well as their knowledge. 
For them it was apparently of less consequence that their predic- 
tions should come true than that they should not let an eclipse take 
place without their having predicted it. 2 

As it is with Egypt that legend associates Thales, it is natural 
to ask whether the Egyptians too were acquainted with the period 
of 223 lunations. We have no direct proof ; but Diodorus Siculus 
says that the priests ,oX. Thebes predicted eclipses quite as well as 
the, Chaldeans, 3 and it is quite possible that tl^e former had learnt 
from the latter the period and the notions on which the successful 
prediction of eclipses depended. It is not, however, essential to 
suppose that Thales got the information from the Egyptians ; he 
may have obtained it more directly. Lydia was an outpost of 

1 Tannery, op. cit., p. 57. 

2 Delambre {Hist, de C astronomie ancienne, i, p. 351) quotes a story that in 
China, in 2159 B.C., the astronomers Hi and Ho were put to death, according 
to law, in consequence of an eclipse of the sun occurring which they had not 
foretold. 

3 Cf. Diodorus, i, c. 50; ii, c. 30. 



^- 



1 8 ΤΗ ALES parti 

Assyrio-Babylonian culture ; this is established by (among other 
things) the fact of the Assyrian protectorate over the kings Gyges 
and Ardys (attested by cuneiform inscriptions) ; and * no doubt the 
inquisitive Ionians who visited the gorgeous capital Sardes, situated 
in their immediate neighbourhood, there first became acquainted 
with the elements of Babylonian science '- 1 

If there happened to be a number of possible solar eclipses in the 
year which (according to Herodotus) Thales fixed, he was not 
taking an undue risk; but it was great luck that it should have 
-been total. 2 

Perhaps I have delayed too long over the story of the eclipse ; 
but it furnishes a convenient starting-point for a consideration 
of the claim of Thales to be credited with the multitude of other 
discoveries in astronomy attributed to him by the Doxographi and 
others. First, did he know the cause of eclipses ? Aetius says 
that he thought the sun was made of an earthy substance, 3 like 
the moon, and was the first to declare that the sun is eclipsed 
when the moon comes in a direct line below it, the image of the 
moon then appearing on the sun's disc as on a mirror ; 4 and again 
he says that Thales, as well as Anaxagoras, Plato, Aristotle, and 
the Stoics, in accord with the mathematicians, held that the moon 
is eclipsed by reason of its falling into the shadow made by the 
earth when the earth is between the two heavenly bodies. 5 But, as 
regards the eclipse of the moon, Thales could not have given this 
explanation, because he held that theearth floated on the water ; 6 from 
which it may also be inferred that he, like his successors down to 
Anaxagoras inclusive, thought the earth to be a disc or a short 




cylinder. And if he had given the true explanation of the solar 
eclipse, it is impossible that all the succeeding Ionian philosophers 
should have exhausted their imaginations in other fanciful explana- 
ί tions such as we find recorded. 7 

We may assume that Thales would regard the sun and the moon 
as discs like the earth, or perhaps as hollow bowls which could 

1 Gomperz, Griechische Denker^ i 3 , p. 421. 2 Tannery, op. cit., p. 60. 

3 Aet. ii. 20. 9 {D. G. p. 349). * Aet. ii. 24. ι (Ζλ G. pp. 353, 354). 

5 Aet. ii. 29. 6 (D. G. p. 360). 

β Theophrastus apud Simpl. in Phys. p. 23. 24 (D. G. p. 475 ; Vors. i 2 , p. 9. 
22); cf. Aristotle, Metaph. A. 3, 983 b 21 ; De caelo ii. 13, 294 a 28. 
7 Tannery, op. cit., p. 56. 






ch.iii THALES 19 

turn so as to show a dark side. 1 We must reject the statements 

of Aeti us that iic was the first to hold that the moon is lit up by y\^^JLx/^<A^ 

the sun, and that it seems to suffer its obscurations each month 

when it approaches the sun, because the sun illuminates it from 

one side only. 2 For it was Anaxagoras who first gave the true 

scientific doctrine that the moon is itself opaque but is lit up by 

the sun, and that this is the explanation no less of the moon's 

phases than of eclipses of the sun and moon ; when we read 

in Theon of Smyrna that, according to Eudemus's History of 

Astronomy, these discoveries were due to Anaximenes, 3 this would 

seem to be an error, because the Doxographi say nothing of any 

explanations of eclipses by Anaximenes, 4 while on the other hand 

Aetius does attribute to him the view that the moon was made "ϊ l *+^ 

of nre,° just as the sun and stars are made of nre.° 

We must reject, so far as Thales is concerned, the traditions that 
'Thales, the Stoics, and their schools, made the earth spherical', 7 -o ^tL•* 

and that 'the school of Thales put the earth in the centre'. 8 
For (1) we have seen that Thales made the earth a circular or 
cylindrical disc floating on the water like a log 9 or a cork; and (2), 
so far as we can judge of his conception of the universe, he would 
appear to have regarded it as a mass of water (that on which the ^ν^^ΑΛ**-^ 
earth floats) with the heavens superposed in the form of a hemisphere 
and also bounded by the primeval water. It follows from this 
conception that for Thales the sun, moon, and stars did not, between ,mm f* m 

^their setting and rising again, continue their circular path below the 
earth, but (as with Anaximenes later) laterally round the earth. 

Tannery 10 compares Thales' view of the world with that found 
in the ancient Egyptian papyri. In the beginning existed the Nu, 
a primordial liquid mass in the limitless depths of which floated 
the germs of things. When the sun began to shine, the earth was 
flattened out and the waters separated into two masses. The one 
gave rise to the rivers and the ocean ; the other, suspended above, 
formed the vault of heaven, the waters above, on which the stars 

1 Tannery, op. cit, p. 70. a Aet. ii. 28. 5 ; 29. 6 (D. G. p. 358. 19 ; p. 360. 16). 

8 Theon of Smyrna, p. 198. 19-199. 2. 
4 Tannery, op. cit., pp. 56, 153. 6 Aet. ii. 25. 2 (D. G. p. 356. 1). 

6 Aet. ii. 20. 2 (D. G. p. 348. 8) ; Hippol. Refnt. i. 7. 4 (D. G. p. 561. 3). 

7 Aet. iii. 10. 1 (D. G. p. 376. 22). * Aet. iii. 11. 1 {D. G. p. 377. 7). 

9 Aristotle, De caelo ii. 13, 294 a 30. 10 Tannery, op. cit., p. 71. 

C 2 



■^^^^-Μ«λ«*^ 



j 



2o THALES part ι 

* and the gods, borne by an eternal current, began to float. The 
sun, standing upright in his sacred barque which had endured 
millions of years, glides slowly, conducted by an army of secondary 
gods, the planets and the fixed stars. The assumption of an 
upper and lower ocean is also old-Babylonian (cf. the division in 
Gen. i. 7 of the waters which were under the firmament from the 
waters which were above the firmament). 

In a passage quoted by Theon of Smyrna, Eudemus attributed 
to Thales the discovery of ' the fact that the period of the sun with 
respect to the solstices is not always the same'. 1 The expres- 
sion is ambiguous, but it must apparently mean the inequality of 
the length of the four astronomical seasons, that is, the four parts 
of the tropical year 2 as divided by the solstices and the equinoxes. 
Eudemus referred presumably to the two written works by Thales 
On the Solstice and On the Equinox? which again would seem to be 
referred to in a later passage of Diogenes Laertius : 'Lobon of Argos 
says that his written works extend to 200 verses '. Now Hesiod, 
in the Works and Days> advises the commencement of certain 
operations, such as sowing, reaping, and threshing, when particular 
constellations rise or set in the morning, and he uses the solstices 
as fixed periods, but does not mention the equinoxes. Tannery 4 
thinks, therefore, that Thales' work supplemented Hesiod's by the 
addition of other data and, in particular, fixed the equinoxes in 
the same way as Hesiod had fixed the solstices. The inequality 
of the intervals between the equinoxes and the solstices in one 
year would thus be apparent. This explanation agrees with the 
remark of Pliny that Thales tixed the matutinal setting of the 
Pleiades on the 25th day from the autumnal equinox. 5 All this 
knowledge Thales probably derived from the Egyptians or the 
Babylonians. The Babylonians, and doubtless the Egyptians also, 

1 Theon of Smyrna, p. 198. 17 (θαλί)? evpe πρώτος) . . . την κατά rets τροπας 
αντον ntpiodov, ώί ουκ "ίση ael συμβαίνει. 

2 The ' tropical year ' is the time required by the sun to return to the same 
position with reference to the equinoctial points, while the ' sidereal year ' is the 
time taken to return to the same position with reference to the fixed stars. 

3 Diog. L. i. 23 ( Vors. i 2 , p. 3. 18). 

4 Tannery, op. cit., p. 66. 

5 Pliny, N. H. xviii, c. 25, § 213 (Vo?s. i 2 , p. 9. 44). This datum points to Egypt 
as the source of Thales' information, for the fact only holds good for Egypt and 
not for Greece (Zeller, i 5 , p. 184 ; cf. Tannery, op. cit., p. 67). 



ch. πι THALES 21 

were certainly capable of determining more or less roughly the 
solstices and the equinoxes; and they would doubtless do this 
fry means of the gnomon, the use of which, with that of the polos, 
the Greeks are said to have learnt from the Babylonians. 1 

Thales equally learnt from the Egyptians his division of the -"~~Tmj^A*>- 
year into 3Q5jJays ; 2 it is possible also that he followed their ~Y^** 
arrangement of months of 30 days each, instead of the practice 
already in his time adopted in Greece of reckoning by lunar months. 

The Doxographi associate Thales with Pythagoras and his school \ 
as having divided the whole sphere of the heaven by five circles, 
the arctic which is always visible, the summer-tropical, the 
equatorial, the winter-tropical, and the antarctic which is always 
invisible ; it is added that the so-called zodiac circle passes obliquely 
to the three middle circles, touching all three, while the meridian V,, 

circle, which goes from north to south, is at right angles to all the ^+*^Ju*£z££~ . 
five circles. 3 But , if Thales had any notion of these circles, it ^tJMt* *****^ 

must have been of the vaguest ; the antarctic circle in particular χήΡ 
presupposes the spherical form for the earth, which was not the X*£" w **^ 

form which Thales gave it. Moreover, the division into zones is 
elsewhere specifically attributed to Parmenides and Pythagoras ; 
and, indeed, Parmenides and Pythagoras were the first to be in 
a position to take this step, 4 as they were the first to hold that 
the earth is spherical in shape. Again, Eudemus is quoted 5 as ) 
distinctly attributing the discovery of the 'cincture of the zodiac / 
(circle) ' to Oenopides, who was at least a century later than Thales.j 

Diogenes Laertius says that, according to some authorities, 
Thales was the first to declare the apparent , size of the sun (and n? 
the moon) to be 1/7 20th part of the circle described by it. 6 The 
version of this story given by Apuleius is worth quoting for a human 
touch which it contains : 

1 Herodotus, ii. 109. 

2 Herodotus (ii. 4) says that the Egyptians were the first of men to discover 
the year, and that they divided it into twelve parts, ' therein adopting a wiser 
system (as it seems to me) than the Greeks, who have to put in an intercalary 
month every third year, in order to keep the seasons right, whereas the Egyptians 
give their twelve months thirty days each and add five every year outside the 
number (of twelve times 30) \ As regards Thales, cf. Diog. L. i. 27 and 24 
(Vors. i 2 , pp. 3. 27; 4. 9). 3 Aet. ii. 12. 1 (D. G. p. 340. 11 sq.). 

4 As to Parmenides cf. Aet. iii. 11. 4 (D. G. p. 377. 18-20). 

6 Theon of Smyrna, p. 198. 14. 

6 Diog. L. i. 24 (Vorsokratiker, i 2 , p. 3. 25). 



22 



ΤΗ ALES part ι 




'The same Thales in his declining years devised a marvellous 
calculation about the sun, which I have not only learnt but verified 
by experiment, showing how often the sun measures by its own 
size the circle which it describes. Thales is said to have communi- 
cated this discovery soon after it was made to Mandrolytus of 
Priene, who was greatly delighted with this new and unexpected 
information and asked Thales to say how much by way of fee he 
required to be paid to him for so important a piece of knowledge. 
"I shall be sufficiently paid", replied the sage, "if, when you set 
to work to tell people what you have learnt from me, you will not 
take credit for it yourself but will name me, rather than another, 
as the discoverer." ' * 

Seeing that in Thales' system the sun and moon did not pass 
under the earth and describe a complete circle, he could hardly 
have stated the result in the precise form in which Diogenes gives 
it. I£ however, he stated its equivalent in some other way, it 
is again pretty certain that he learnt it from the Egyptians or 
Babylonians. Cleomedes, 2 indeed, says that, by means of a water- 
clock, we can compare the water which flows out during the time 
that it takes the sun when rising to get just clear of the horizon 
with the amount which flows out in the whole day and night ; 
in this way we get a ratio of ι to 750; and he adds that this 
method is said to have been first devised by the Egyptians. Again, 
it has been suggested 3 that the Babylonians had already, some 
sixteen centuries before Christ, observed that the sun takes i/30th 
of an hour to rise. This would, on the assumption of 24 hours for 
a whole day and night, give for the sun's apparent diameter 1/7 20th 
of its circle, the same excellent approximation as that attributed 
to Thales. But there is the difficulty that, when the Babylonians 
spoke of i/3oth of an hour in an equinoctial day as being the 
1 measure ' (opos) of the sun's course, they presumably meant i/30th 
of their doub/e-hour, of which there are 12 in a day and night, 
so that, even if we assume that the measurement of the sun's 
apparent diameter was what they meant by opos, the equivalent 

1 Apuleius, Flor. 18 {Vors. i 2 , p. 10. 3- 11). 

8 Cleomedes, De motu circulari corporum caelestium ii. 1, pp. 136. 25-138. 
6, ed. Ziegler. 

3 Hultsch, Poseidonios iiber die Grbsse und Entfernung der Sonne, 1897, 
pp. 41 , 42. Hultsch quotes Achilles, hagoge in Aratiphaen. \%(Uranolog. Petavii, 
Paris, 1630, p. 137); Brandis, Miinz-, Mass- und Gewichtswesen in Vorderasien, 
p. 17 sq. ; Bilfinger, Die babylonische Doppelstunde, Stuttgart, 1888, p. 21 sq. 
The passage of Achilles is quoted in extenso by Bilfinger, p. 21. 



ch.iii THALES 23 

would be i°, not J° as Hultsch supposes. 1 However, it is difficult to 
believe that Thales could have made the estimate of 1/7 20th of 
the sun's circle known to the Greeks ; if he had, it would be very- 
strange that it should have been mentioned by no one earlier than 
Archimedes, and that Aristarchus should in the first instance have 
used the grossly excessive value of 2° which he gives as the angular 
diameter of the sun and moon in his treatise On the sizes and 
distances of the sun and moon, and should have been left to dis- 
cover the value of -|° for himself as Archimedes says he did. 2 

A few more details of Thales' astronomy are handed down. He 
said of the Hyades that there are two, one north and the other 
south. 3 According to Callimachus, 4 he observed the Little Bear ; 
'Jie was said to have used as a standard [i. e. for finding the pole] 
the small stars of the Wain, that being the method by which 
Ehoenician navigators steer their course.' According to Aratus 5 
the Greeks sailed by the Great Bear, the Phoenicians by the Little 
Bear. Consequently it would seem that Thales advised the Greeks 
to follow the Phoenician plan in preference to their own. This use 
of the Little Bear was probably noted in the handbook under the 
title of Nautical Astronomy attributed by some to Thales, and 
by others to Phocus of Samos, 6 which was no doubt intended to 
improve upon the Astronomy in poetical form attributed to Hesiod, 
as in its turn it was followed by the Astrology of Cleostratus. 7 

1 An estimate amounting to i° is actually on record in Cleomedes (De motu 
circulari, ii. 3, p. 172. 25, Ziegler), who says that ' the size of the sun and moon 
alike appears to our perception as 12 dactylV. Though this way of describing 
the angle follows the Babylonian method of expressing angular distances 
between stars in terms of the ell {πήχνς) consisting of 24 dactyli and equivalent 
to 2°, it does not follow that the estimate itself is Babylonian. For the same 
system of expressing angles may have been used by Pytheas and was certainly 
used by Hipparchus (cf. Strabo, ii. 1. 18, p. 75 Cas., Hipparchiin Arati et Eudoxi 
phaenomena coimnent. ii. 5. I, p. 186. II, Manit., and Ptolemy, Syntaxis vii. 1, 
vol. ϋ,ρρ. 4-8, Heib.). 

2 Archimedes, ed. Heiberg, vol. ii, p. 248. 19 ; The Works oj Archimedes, ed. 
Heath, p. 223. 

3 Schol. Aral. 172, p. 369. 24 (Vors. ii. I 2 , p. 652). 

4 In Diog. L. i. 23 ( Vors. i 2 , p. 3. 14 ; cf. ii. 2, p. v). 

5 Aratus, lines 27, 37-39 ; cf. Ovid, Tristia iv. 3. 1-2 : 

1 Magna minorque ferae, quarum regis altera Graias, 
Altera Sidonias, utraque sicca, rates ' ; 

Theo in Arati phaen. 27. 39 : Scholiast on Plat. Rep. 600 A. 

6 Diog. L. i, p. 23 ; Simpl. in Phys.p. 23. 29 ; Plutarch, Pyth. or. 18, 402 f ( Vors. 
i 2 , pp. 3. 12 ; 11. 7, 13). 7 Diels, Vors. ii. I 2 , p. 652 ; cf. pp. 499, 502. 



IV 
ANAXIMANDER 

ANAXIMANDER of Miletus (born probably in 611/10, died soon 
after 547/6 B.C.), son of Praxiades, was a fellow citizen of Thales, 
with whom he was doubtless associated as a friend if not as a pupil. 
A remarkably original thinker, Anaximander may be regarded as 
the father or founder of Greek, and therefore of western, philosophy. 
He was the first Greek philosopher, so far as is known, who 
ventured to put forward his views in a formal written treatise. 1 
This was a work About Nature? though possibly that title was 
given to it, not by Anaximander himself, but only by later writers. 3 
The amount of thought which went to its composition and the 
maturity of the views stated in it are indicated by the fact that 
it was not till the age of 64 that he gave it to the world. 4 The 
work itself is lost, except for a few lines amounting in no case 
to a complete sentence. 

Anaximander boldly maintained that the earth is in the centre 
of the universe, suspended freely and without support, 5 whereas 
Thales regarded it as resting on the water, and Anaximenes as 
supported by the air. It remains in its position, says Anaximander, 
because it is at an equal distance from all the rest (of the heavenly 
bodies). 6 Aristotle expands the explanation thus : 7 ' for that 
which is located in the centre and is similarly situated with 
reference to the extremities can no more suitably move up than 

1 Themistius, Orationes, 36, p. 317 C (Vors. i 2 , p. 12. 43). 

2 Ibid. ; Suidas, s. v. 

I Zeller, Philosophie der Griechen, i 5 , p. 197. 

4 Diog. L. ii. 2 (Vors. i 2 , p. 12. 7-10). 

5 Hippol. Refut. i. 6. 3 {D. G. p. 559. 22 ; Vors, i 2 , p. 14. 5). 

6 Ibid. ; cf. Plato's similar view vaPhaedo 108 E-109 A. 

7 De caelo ii. 13, 295 b 10-16. It is true that Eudemus (in Theon of Smyrna, 
p. 198. 18) is quoted as saying that Anaximander held that 'the earth is suspended 
freely and moves (κινΰται) about the centre of the universe ' ; but there must 
clearly be some mistake here ; perhaps κΜΪται should be κείται (' lies '). 



ANAXIMANDER f 25 

down or laterally, and it is impossible that it should move in 
opposite directions (at the same time), so that it must necessarily 
remain at rest.' Aristotle admits that the hypothesis is daring 
and brilliant, but argues that it is not true : one of his grounds 
is amusing, namely, that on this showing a hungry and thirsty man 
with food and wine disposed at equal distances all round him would 
have to starve because there would be no reason for him to stretch 
his hand in one direction rather than another * (presumably the first 
occurrence of the well-known dilemma familiar to the schoolmen 
as the * Ass of Buridan '). 

According to Anaximander, the earth has the shape of a cylinder, 
round, * like a stone pillar ' ; 2 one of its two plane faces is that on ) 
which we stand, the other is opposite ; 3 its depth, moreover, is one- 
third of its breadth. 4 

Still more original is Anaximander's conception of the origin and 
substance of the sun, moon, and stars, and of their motion. As 
there is considerable difference of opinion upon the details of the 
system, it will be well, first of all, to quote the original authorities, 
beginning with the accounts of the cosmogony. 



* Anaximander of Miletus, son of Praxiades, who was the successor 



and pupil of Thales, said that the first principle (i.e. material cause) 

and element of existing things is the. Infinite, and he was the first (^^ΊΓ 

to introduce this name for the first principle. He maintains that «*a 

it is neither water nor any other of the so-called elements, but 

another sort of substance, which is infinite, and from which 

all the heavens and the worlds in them are produced ; and into 

that from which existent things arise they pass away once more, 

" as is ordained ; for they must pay the penalty and make reparation 

to one another for the injustice they have committed, according to 

the sequence of time", as he says in these somewhat poetical 

terms.' 5 

1 Aristotle, De caelo ii. 13, 295 b 32. 

2 Hippol. Refut. i. 6. 3 {D. G. p. 559. 24 ; Vors. i 2 , p. 14. 6) ; Aet. iii. 10. 2 
{D. G. p. 376; Vors. i 8 , p. 16. 34). 

8 Hippol., loc. cit. 

4 Ps. Plut. Stromat. 2 (D. G. p. 579. 12; Vors. \ 2 , p. 13. 34). 

5 Simplicius, in Phys. p. 24. 13 {Vors. i 2 , p. 13.2-9). The passage is from 
Theophrastus's Phys. Ofiin., and the words in inverted commas at all events are 
Anaximander's own. I follow Burnet {Early Greek Philosophy, p. 54) in making 
the quotation begin at 'as is ordained'; Diels includes in it the words just 
preceding 'and into that from which . . . .' 




rl 



•" 



26 ANAXIMANDER part ι 

1 Anaximander said that the Infinite contains the whole cause of 
the generation and destruction of the All ; it is from the Infinite 
that the heavens are separated off, and generally all the worlds, 
which are infinite in number. He declared that destruction and, 
long before that, generation came about for all the worlds, which 
arise in endless cycles from infinitely distant ages.' * 

4 He says that this substance [the Infinite] is eternal and ageless, 
and embraces all the worlds. And in speaking of time he has in 
mind the separate (periods covered by the) three states of coming 
into being, existence, and passing away.' 2 

' Besides this (Infinite) he says there is an eternal motion, in the 
course of which the heavens are found to come into being.' 3 

'Anaximander says eternal motion is a principle older than 
the moist, and it is by this eternal motion that some things are 
β/^ 1 generated and others destroyed.' 4 

c He says that (the first principle or material cause) is boundless, 
in order that the process of coming into being which is set up may 
not suffer any check.' 5 

'Anaximander was the first to. assume the Infinite as first 
principle in order that he may have it available for his new births 
without stint.' 6 

J Anaximander . . . said that the world is perishable/ 7 

' Those who assumed that the worlds are infinite in number, as 
did Anaximander, Leucippus, Democritus, and, in later days, 
Epicurus, assumed that they also came into being and passed 
away, ad infi7iitum, there being always some worlds coming into 
being and others passing away ; and they maintained that motion 
is eternal; for without motion there is no coming into being or 
passing away.' 8 

' Anaximander says that that which is capable of begetting the 
hot and the cold out of the eternal was separated off during the 
coming into being of our world, and from the flame thus produced 
a sort of sphere was made which grew round the air about the 
earth as the bark round the tree ; then this sphere was torn off and 

1 Ps. Plut. Stromal. 2 (D. G. p. 579 ; Vors. i 2 , p. 13. 29 sq.). This passage 
again is from Theophrastus. 

2 Hippol. Refut. i. 6. 1 (D. G. p. 559 ; Vors. i 2 , pp. 13. 44-14. 2). 

3 Ibid. i. 6. 2. 

4 Hermias, Irris. 10 (D. G. p. 653 ; Vors. i 2 , p. 14. 21). 

5 Aet. i. 3. 3 (D. G. p. 277 ; Vors. i 2 , p. 14. 29). 

6 Simplicius on De caeio, p. 615. 13 (Vors. i 2 , p. 15. 24). In this passage 
Simplicius calls Anaximander a ' fellow citizen and friend ' of Thales (θαλον 
πολίτης και eraipos) ; these appear to be the terms used by Theophrastus, to 
judge by Cicero's equivalent 'popularis et sodalis ' (Acad. pr. ii. 37. 118). 

7 Aet. ii. 4. 6 (D. G. p. 331 ; Vors. i 2 , p. 15. 33). 

8 Simplicius, in Phys. p. 1121. 5 (Vors. i 2 , p. 15. 34-8). 




CH. iv ANAXIMANDER 27 

became enclosed in certain circles or rings, and thus were formed 
the sun, the moon, and the stars.' x 

* The stars are produced as a circle of fire, separated off from the 
fire in the universe and enclosed by air. They have as vents certain 
pipe-shaped passages at which the stars are seen ; it follows that 
it is when the vents are stopped up that eclipses take place.' 2 

1 The stars are compressed portions of air, in the shape of wheels, 
filled with fire, and they emit flames at some point from small 
openings.' 3 

1 The moon sometimes appears as waxing, sometimes as waning, 
to an extent corresponding to the closing or opening of the 
passages.' 4 

Further particulars are given of the circles of the sun and moon, 
including the first speculation about their sizes: 

' The sun is a circle 28 times the size of the earth ; it is like 
a wheel of a chariot the rim of which is hollow and full of fire, 
and lets the fire shine out at a certain point in it through an 
opening like the tube of a blow-pipe ; such is the sun.' 5 

' The stars are borne by the circles and the spheres on which 
each (of them) stands.' 6 

1 Ps. Plut. Stromat. loc. cit. 

2 Hippol. Refut. i.6. 4 (D. G. pp. 559, 560; Vors. i 2 , p. 14. 8). 

3 Aet. ii. 13. 7 (D. G. p. 342 ; Vors. i*, p. 15. 39). 

4 Hippol., loc. cit. 

5 Aet. ii. 20. 1 {D. G. p. 348 ; Vors. i 2 , p. 16. 8). 

6 Aet. ii. 16. 5 (D.G. p. 345 ; Vors. i 2 , p. 15. 43. This sentence presents diffi- 
culties. It occurs in a collection of passages headed * Concerning the motion of 
Stars ', and reads thus : 'Αναξίμανδρος νπό των κύκλων κα\ των σφαιρών, εφ' ων 
έκαστος βεβηκε, φερεσθαι. If έκαστος means έκαστος τών αστέρων, each of the 
stars, the expression εφ' ων έκαστος βεβηκε, ' on which each of them stands ' or 
1 is fixed ', is certainly altogether inappropriate to Anaximander's system ; it 
suggests Anaximenes' system of stars * fixed like nails on a crystal sphere '; I am 
therefore somewhat inclined to suspect, with Ν euhiiuser (Anaximander Milesius, 
p. 362 note), that the words ε'φ' ων έκαστος βεβηκε (if not και τών σφαιρών also) 
are wrongly transferred from later theories to that of Anaximander. It occurred 
to me whether έκαστος could be έκαστος τών κύκλων, i each of the circles ' ; for it 
would be possible, I think, to regard the circles as 'standing' or 'being fixed' 
on (imaginary) spheres in order to enable them to revolve about the axis of such 
spheres, it being difficult to suppose a wheel to revolve about its centre when it 
has no spokes to connect the centre with the circumference. 

Diels (' Ueber Anaximanders Kosmos' in Archiv filr Gesch. d. Philosophic, x, 
1897, p. 229) suggests that we may infer from the word ' spheres ' here used that the 
rings are not separate for each star, but that the fixed stars shine through vents 
on one ring (which is therefore a sphere) ; the planets with their different motions 
would naturally be separate from this. I doubt, however, whether this is 
correct, since all the rings are supposed to be like wheels!; they are certainly 
not spheres. But no doubt the Milky Way may be one ring from which 



\ /.,A-4a. ■- ' 



2 8 ANAXIMANDER part ι 

' The circle of the sun is 27 times as large (as the earth and that) 
of the moon (is 19 times as large as the earth).' x 

1 The sun is equal to the earth, and the circle from which the sun 
gets its vent and by which it is borne round is 27 times the size of 
the earth.' 2 

' The eclipses of the sun occur through the opening by which the 
fire finds vent being shut up.' 3 

'The moon is a circle 19 times as large as the earth; it is 
similar to a chariot- wheel the rim of which is hollow and full 
of fire, like the circle of the sun, and it is placed obliquely like the 
other ; it has one vent like the tube of a blowpipe ; the eclipses of 
the moon depend on the turnings of the wheel.' * 

' The moon is eclipsed when the opening in the rim of the wheel 
is stopped up.' 5 

4 The sun is placed highest of all, after it the moon, and under 
them the fixed stars and the planets.' 6 

We are now in a position to make some comments. First, what 
is the nature of the eternal motion which is an older principle than 
water and by which some things are generated and others destroyed ? 
Teichmliller held it to be circular revolution of the Infinite, which 
he supposed to be a sphere, about its axis ; 7 Tannery adopted the 
same view. 8 Zeller 9 rejects this for several reasons. There is no 
evidence that Anaximander conceived the spherical envelope of 
fire to be separated off by revolution of the Infinite and spread 
out over the surface of its mass ; the spherical envelope lay, not 
round the Infinite, but round the atmosphere of the earth, and it 
was only the world, when separated off, which revolved ; it is the 
world too, not the Infinite, which stretches at equal distances, and 
therefore in the shape of a sphere, round the earth as centre. 
Lastly, a spherical Infinite is in itself a gross and glaring contra- 
diction, which we could not attribute to Anaximander without 

a multitude of stars flame forth at different vents : this may indeed be the idea 
from which the whole theory started (Tannery, op. cit., p. 91 ; Burnet, Early 
Greek Philosophy, p. 69). 

1 Hippol., Refut. i. 6. 5 {D. G. p. 560 ; Vors. i 2 , p. 14. 12, and ii. I 2 , p. 653). 

2 Aet. ii. 21. 1 (D. G. p. 351 ; Vors. i 2 , p. 16. 11). 

3 Aet. ii. 24. 2 {D. G. p. 354 ; Vors. i 2 , p. 16. 13). 

4 Aet. ii. 25. 1 (D. G. p. 355 ; Vors. i 2 , p. 16. 15). 

5 Aet. ii. 29. 1 (D. G. p. 359; Vors. i 2 , p. 16. 19). 
* Aet. ii. 15. 6 {D. G. p. 345 ; Vors. i 2 , p. 15. 41). 

7 Teichmiiller, Studien zur Gesch. der Begriffe, Berlin, 1874, pp. 25 sqq. 

8 Tannery, op. cit., pp. 88 sqq. 
s Zeller, i 5 , p. 221. 



ch.iv ANAXIMANDER 29 

direct evidence. Tannery 1 gets over the latter difficulty by the 

assumption that the Infinite was not something infinitely extended 

in space but qualitatively indeterminate only, and in fact finite in 

extension. This is rather an unnatural interpretation, especially 

in view of what we are told of the 'infinite worlds' which arise 

from the Infinite substance. The idea here seems to be that the 

Infinite is a boundless stock from which the waste of existence is P^pf 4, 

continually made good. 2 With regard to the 'infinite worlds^ 

Zeller 8 held that they were an infinity of successive worlds, not -Ji&**°~ ' 

an unlimited number of worlds existing, or which may exist, at AA. 

the same time, though of course all are perishable ; but in order 

to sustain this view Zeller was obliged to reject a good deal of the 

evidence. Burnet 4 has examined the evidence afresh, and adopts 

the other view. In particular, he observes that it would be 

very unnatural to understand the statement that the Boundless 

'encompasses the worlds' of worlds succeeding one another in 

time ; for on this view there is at a given time only one world 

to ' encompass '. Again, when Cicero says Anaximander's opinion 

was that there were gods who came into being, rising and setting 

at long intervals, and that these were the 'innumerable worlds' 5 (cf. 

Aetius's statement that, according to Anaximander,the 'innumerable 

heavens' were gods 6 ), it is more natural to take the long intervals 

as intervals of space than as intervals of time ; 7 and, whether this 

is so or not, we are distinctly told in a passage of Stobaeus that 

'of those who declared the worlds to be infinite in number, 

Anaximander said that they were at equal distances from one 

another', a passage which certainly comes from Aetius. 8 Neu- 

hauser, 9 too, maintains that Anaximander asserted the infinity of 

worlds in two senses, holding both that there are innumerable 

worlds co-existing at one time and separated by equal distances, 

and that these worlds are for ever, at certain (long) intervals of 

1 Tannery, op. cit., pp. 146, 147. 

2 Burnet, Early Greek Philosophy ', p. 55. 

3 Zeller, i 5 , pp. 229-36. 

4 Burnet, Early Greek Philosophy, pp. 62-6. 

5 Cicero, De nat. deor. i. 10. 25 ( Vors. i 2 , p. 15. 27). 

6 Aet. i. 7. 12 (D. G. p. 302 ; Vors. i 2 , p. 15. 26). 

7 Probably, as Burnet says, Cicero found διαστήμασιν in his Epicurean source. 

8 Aet. ii. 1. 8 (D. G. p. 329 ; Vors. i 2 , p. 15. 32). 

9 Neuhauser, Anaximander Milesius, pp. 327-35. 



3° 



ANAXIMANDER 



PARTI 






y 



time, 1 passing away into the primordial Infinite, and others con- 
tinually succeeding to their places. 2 

The eternal motion of the Infinite would appear to have been 
the * separating-out of opposites \ 3 but in what way this operated 
is not clear. The term suggests some process of shaking and 
sifting as in a sieve. 4 Neuhauser 5 holds that it is not spatial 
motion at all, but motion in another of the four Aristotelian senses, 
namely generation, which takes the form of the * separating-out of 
opposites ', condensation and rarefaction incidentally playing a part 
in the process. 

As regards the motion by which the actual condition of the 
world was brought about (the earth in the centre in the form of 
a flat cylinder, the sun, moon, and stars at different distances from 
the earth, and the heavenly bodies revolving about the axis of the 
universe), Neuhauser 6 maintains that it was the motion of a vortex 
such as was assumed by Anaxagoras, the earth being formed in 
the centre by virtue of the tendency of the heaviest of the things 
whirled round in a vortex to collect in the centre. But there is 
no evidence of the assumption of a vortex by Anaximander; 
Neuhauser relies on a single passage of Aristotle, which however 
does not justify the inference drawn from it. 7 

1 κατά την του χρόνου τάξιν, Simpl. in Phys. p. 24. 20 ( Vors. i 2 , p. 13. 9). 

2 Cf. Simpl. in Phys. p. 1121. 5 (Vors. i 2 , p. 15. 34-8, quoted above, p. 26). 

s oi Se €Κ του evos (νούσας τας βναντιότηταε 4κκρίν(σθαΐ, ωσπερ Αναξίμανδρος φησι, 
Aristotle, Phys. i. 4, 187 a 20. 

4 Burnet, Early Greek Philosophy, p. 61. 

5 Neuhauser, Anaxi7nander Milesius, pp. 305-15. 

6 Neuhauser, Anaximander Milesius, pp. 409-21. 

7 The passage is Aristotle, De caelo ii. 13, 295 a 9 sqq. It is there stated that 
' if the earth, as things are, is kept by force where it is, it must also have come 
together (by force) through being carried towards the centre by reason of the 
whirling motion ; for this is the cause assumed by everybody on the ground of 
what happens in fluids and with reference to the air, where the bigger and the 
heavier things are always carried towards the middle of the vortex. Hence it is 
that all who describe the coming into being of the heaven say that the earth came 
together at the centre ; but the cause of its remaining fixed is still the subject 
of speculation. Some hold . . .' Now Neuhauser paraphrases the passage thus : 
'AH philosophers who hold that the world was generated or brought into being 
maintain that the earth is not only kept by force in the middle of the world, but 
was, at the beginning, also brought together by force. For all assign as the 
efficient cause of the concentration of the earth in the middle of the world a 
vortex (δίνη), arguing from what happens in vortices in water or air.' It is clear 
that Aristotle says no such thing. He says that the philosophers referred to 
assert that the earth comes together at the centre, but not that they hold that it 
is kept there by force ; indeed he expressly says later (295 b 10-16) that Anaxi- 



CH.IV ANAXIMANDER 31 

We come now to Anaximander's theory of the sun, moon, and 
stars. The idea of the formation of tubes of compressed air within 
which the fire of each star is shut up except for the one opening 
is not unlike Laplace's hypothesis with reference to the origin of 
Saturn's rings. 1 A question arises as to how, if rings constituting 
the stars are nearer than the circles of the sun and moon, they fail 
to obstruct the light of the latter. Tannery 2 suggests that, while 
of course the envelopes of air need not be opaque, the rarefied 
fluid within the hoops, although called by the name of fire, may 
also be transparent, and not be seen as flame except on emerging 
at the opening. The idea that the stars are like gas-jets, as it- 
were, burning at holes in transparent tubes made of compressed fMU 
air is a sufficiently original conception. 

But the question next arises, in what position do the circles, 
wheels, or hoops carrying the sun, moon, and stars respectively 
revolve about the earth? Zeller and Tannery speak of them as 
' concentric \ their centres being presumably the same as the centre 
of the earth ; and there is nothing in the texts to suggest any other 
supposition. The hoops carrying the sun and moon ' lie obliquely ', 
this being no doubt an attempt to explain, in addition to the daily 
rotation, the annual movement of the sun and the monthly move- 
ment of the moon. Tannery raises the question of the heights 
(' hauteurs ') of these particular hoops, by which he seems to mean 
their breadths as they would be seen (if visible) from the centre. 
Thus, if the bore of the sun's tube were not circular but flattened 
(like a hoop), in the surface which it presents towards the earth, 
to several times the breadth of the sun's disc, it might be possible 
to explain the annual motion of the sun by supposing the opening 
through which the sun is seen to change its position continually 
on the surface of the hoop. But there is nothing in the texts to 
support this. Zeller 3 feels difficulty in accepting the sizes of the 
hoops as given, on the supposition that the earth is the centre. 

mander regarded the earth as remaining at the centre without any force to keep 
it there. Again ' everybody ' is not ' all philosophers ', but ' people in general '. 
Lastly, the tendency of the heavier things in a vortex to collect at the centre 
might easily suggest that the earth had come together in the centre because it 
was heavy, without its being supposed that a vortex was the only thing that could 
cause it to come together. 

1 Tannery, op. cit., p. 88. a Ibid. p. 92. 

3 Zeller, i 5 , pp. 224, 225. 



3* 



ANAXIMANDER part ι 



For we are told that the sun's circle or wheel is 27 or 28 times 
the size of the earth, while the sun itself is the same size as the 
earth ; this would mean that the apparent diameter of the sun's 
disc would be a fraction of the whole circumference of the ring 
represented by 1/2 8 π, that is, the angular diameter would be about 
36o°/88, or a little over 4 , which is eight times too large, and 
would be too great an exaggeration to pass muster even in those 
times. Zeller therefore wonders whether perhaps the sun's circle 
should be 27 times the moon's circle, which would make it 513 
times the size of the earth. But the texts, when combined, are 
against this, and further it would make the apparent diameter of 
the sun much too small. According to Anaximander, the sun 
itself is of the same size as the earth ; therefore, assuming d to 
be the diameter of the sun's disc and also the diameter of the earth, 
the circumference of the sun's hoop would be ^i^nd, so that the 
apparent diameter of the sun would be about i/i6ooth part of its 
circle, or less than half what it really is. Teichmiiller 1 and 
Neuhauser 2 try to increase the size of the sun's hoop 3-1416 times, 
apparently by taking the diameter of the hoop to be 28 times 
the circumference of the earth, ' because the measurement clearly 
depended on an unrolling ' ; but this is hardly admissible ; the 
texts must clearly be comparing like with like. Sartorius s feels 
the same difficulty, and has a very interesting hypothesis designed 
to include provision for the sun's motion in the ecliptic as well 
as the diurnal rotation. He bases himself on a passage of Aristotle 
which, according to a statement of Alexander Aphrodisiensis made 
on the authority of Theophrastus, refers to Anaximander's system. 
Aristotle speaks of those who explain the sea by saying that 

' at first all the space about the earth was moist, and then, as it 
was dried up by the sun, one portion evaporated and set up winds 
and the turnings (τροπαί) of the sun and moon, while the remainder 
formed the sea ' ; 4 

1 Teichmiiller, Studien zur Geschichte der Begriffe, 1874, pp. 16, 17. 

2 Neuhauser, Anaximander Μ ilesins, p. 371. 

3 Sartorius, Die Entwicklung der Astronomie beiden Griechen bis Anaxagoras 
und Empedokles, pp. 29, 30. 

4 Aristotle, Meteorologica ii. 1, 353 b 6-9. A note of Alexander (in Meteor. 
p. 67. 3 ; see D.G. p. 494; Vors. i 2 , p. 16. 45) explains the passage thus: 'For, the 
space round the earth being moist, part of the moisture is then evaporated by 
the sun, and from this arise winds and the turnings of the sun and moon, the 



ch. iv ANAXIMANDER 33 

and again he says in another place : 

1 The same absurdity also confronts those who say that the earth, 
too, was originally moist, and that, when the portion of the world 
immediately surrounding the earth was warmed by the sun, air was 
produced and the whole heaven was thus increased, and that this is 
how winds were caused and the turnings of the heaven brought 
about.' 1 

It is on these passages that Zeller 2 grounds his view that the \ 
heavens are moved by these winds {πνεύματα) and not by the 
eternal rotational movement of the Infinite about its axis assumed 
by Teichmuller and Tannery ; accordingly, Zeller cannot admit 
that the word τροπαί in these passages is used in its technical sense 
of ' solstices \ 3 Sartorius, however, clearly takes the τροπαί to refer 
specially to the solstices (so does Neuhauser 4 ), and he shows how 
the motions of the sun could be represented by two different but 
simultaneous revolutions of the sun's wheel or hoop. Suppose the 
wheel to move bodily in such a way that (i) its centre describes 
a circle in the plane of the equator, the centre of which is the 
centre of the earth, while (2) the plane of the wheel is always 
at right angles to the plane of the aforesaid circle, and always 
touches its circumference ; lastly, suppose the wheel to turn about 



meaning being that it is by reason of these vapours and exhalations that the 
sun and moon execute their turnings, since they turn in the regions where they 
receive abundant supplies of this moisture ; but the part of the moisture which is 
left in the hollow places (of the earth) is the sea.' 

1 Aristotle, Meteorologica ii. 2, 355 a 21. 2 Zeller, i 6 , p. 223. 

3 Zeller (i 5 , pp. 223, 224) has a note on the meanings of the word τροπή. Even 
in Aristotle it does not mean 'solstice' exclusively, because he speaks of 'τροπαί 
of the stars ' (De caelo ii. 14, 296 b 4), ' τροπαί of the sun and ?noon ' (Meteor, ii. 
I, 353 b 8), and 'τροπαί of the heaven' (according to the natural meaning of τας 
τροπας αυτοί), 355 a 25). It is true that τροπαί could be used of the moon in 
a sense sufficiently parallel to its use for the solstices, for, as Dreyer says 
(Planetary Systems, p. 17, note 1), the inclination of the lunar orbit to that of 
the sun is so small (5 ) that the phenomena of 'turning-back' of sun and moon 
are very similar. But the use of the word by Aristotle with reference to the 
stars and the heaven shows that it need not mean anything more than the 
1 turnings ' or revolutions of the different heavenly bodies. Zeller's view is, I 
think, strongly supported by a passage in which Anaximenes is made to speak of 
stars 'executing their turnings' (τροπαί ποιύσβαι Aet. ii. 23. 1, D. G. p. 352) and 
the passage in which Anaximander himself is made to say that the eclipses of 
the moon depend on 'the turnings (τροπαί) of its wheel' (Aet. ii. 25. 1, D. G. 
p. 355 b 22). 

4 Neuhauser, op. cit., p. 403. 

1410 D 



34 



ANAXIMANDER 



PARTI 



its own centre at such speed that the opening representing the sun 
completes one revolution about the centre of the wheel in a year, 
and suppose the centre of the wheel to describe the circle in the 
plane of the equator at uniform speed in one day. 

In the figure appended, Ε represents the earth, the C's are posi- 
tions of the centre of the sun's hoop or wheel ; 

5 α represents the sun's position at the vernal equinox ; 
c M fi „ „ summer solstice ; 






autumnal equinox ; 
winter solstice. 



North Pole 




Sourb Role, 



Fig. 2. 



At the winter solstice the sun is south of the equator, at the 
summer solstice north of it, and the diameter of the wheel corresponds 
to an angle at Ε which is double of the obliquity of the ecliptic, 
say 47 . Now, as the diameter of the sun's wheel is 28 times the 
diameter of the earth, i.e. of the sun itself (which is the same size 
as the earth), the angular diameter of the sun at Ε will be about 
47°/28 or i°4i / . This is still far enough from the real approximate 
value §°, but it is much nearer than the 4 obtained from the 
hypothesis of a hoop with its centre at the centre of the earth. 



CH. iv ANAXIMANDER $$ 

Let us consider what would be the distance of the sun from the 
earth on the assumption that the sun's diameter (supposed to be 
equal to that of the earth) subtends at Ε an angle of ι|°. Ιϊ d 
be the diameter of the earth, and D the distance of the sun from 
the earth, we shall have approximately 

36o^/i| = 2wD, 
or D = 34«4 times the diameter of the earth. 

But Sartorius's hypothesis is nothing more than an ingenious 
guess, as the texts give no colour to the idea that Anaximander 



Fig. 3- 

intended to assign a double motion to the sun, nor is there anything 
to suggest that the hoops of the sun and moon moved in any 
different way from those of the stars, except that they were both 
* placed obliquely \ 

The hypothesis of concentric rings with centres at the centre of 
the earth seems therefore to be the simplest. 

Neuhauser, 1 in his attempted explanation of Anaximander's theory 
of the sun's motion, contrives to give to τροπαϊ ηλίου the technical 
meaning of solstices, while keeping the ring concentric with the 
earth. The flat cylinder (centre O) is the earth, N.P. and S.P. are 
the north and south poles, the equator is the circle about A A' as 

1 Neuhauser, pp. 405-8 and Fig. 2 at end. 
D % 



3* 



ANAXIMANDER 



PARTI 




diameter and perpendicular to the plane of the paper. Neuhauser 
then supposes the plane of the sun's circle or hoop to be differently 
inclined to the circle of the equator at different times of the year, 
making with it at the summer solstice and at the winter solstice 
angles equal to the obliquity of the ecliptic in the manner shown in 
the figure, where the circle on AA' as diameter in the plane of the 
paper is the meridian circle and SS' is the diameter of the sun's 
ring at the summer solstice, BB' the diameter of the sun's ring at 
the winter solstice. Between the extreme positions at the solstices 
the plane of the sun's hoop changes its inclination slightly day by 
day, its section with the meridian plane moving gradually during 
one half of the year from the position SS' to the position BB', and 
during the other half of the year from BB' back to SS'. As it 
approaches the summer-solstitial position, it is prevented from 
swinging further by the winds, which are caused by exhalations, and 
which by their pressure on the sun's ring force it to swing back again. 
The exhalations and winds only arise in the regions where there is 
abundant water. Neuhauser supposes that Anaximander had the 
Mediterranean and the Black Sea in mind, and that their positions 
sufficiently ' correspond ' (?) to the summer-solstitial position SS' to 
enable the winds to act as described. There is no sea in such 
a position as would enable winds arising from it to repel the sun's 
ring in the reverse direction from BB' to SS'-, consequently 
Neuhauser has to suppose that the ring has an automatic tendency 
to swing towards the position SS' and that it begins to go back 
from BB', of itself, as soon as the force of the wind which repelled 
it from SS' ceases to operate. There is, however, no evidence in 
the texts to confirm in its details this explanation of the working of 
Anaximander's system ; on the contrary, there seems to be positive 
evidence against it in the phrase ' lying obliquely ', used of the hoops 
of the sun and moon, which suggests that the hoops remain at fixed 
inclinations to the plane of the equator instead of oscillating, as 
Neuhauser 's theory requires, between two extreme positions rela- 
tively to the equator. 

In any case Anaximander's system represented an enormous 
advance in comparison with those of the other Ionian philosophers 
in that it made the sun, moon, and stars describe circles, passing 
right under the earth (which was freely suspended in the middle), 



CH.iv ANAXIMANDER 3 y 

instead of moving laterally round from the place of setting to the 
place of rising again. 

We are told by Simplicius that 

1 Anaximander was the first to broach the subject of sizes and 
distances ; this we learn from Eudemus, who however refers to the 
Pythagoreans the first statement of the order (of the planets) in 
space.' * 

This brings us back to the question of the sizes of the hoops of 
the sun and moon as given by Anaximander. We observe that in 
one passage the sun's circle is said to be 28 times as large as the 
earth, while in another the circle ' from which it gets its vent ' is 
27 times as large as the earth. Now, on the hypothesis of 
concentric rings, we, being in the centre, of course see the inner 
circumference at the place where the sun shines through, the 
sun's light falling, like a spoke of the wheel, towards the centre. 
The words, then, used in the second passage, referring to the circle 
from which the sun gets its vent, suggest that the ' 27 times ' refers 
to the inner circumference of the wheel, while the ' 28 times ' refers 
to the outer; 2 the breadth therefore of the sun's wheel measured 
in the direction from centre to circumference is equal to once the 
diameter of the earth. A like consideration suggests that it is the 
outer circumference of the moon's hoop which is 19 times the size 
of the earth, and that the inner circumference is 18 times the size 
of the earth ; nothing is said in our texts about the size of the moon 
itself. Nor are we told the size of the hoops from which the stars 
shine, but, as they are in Anaximander's view nearer to the earth 

1 Simplicius on De caelo, p. 471. 4, ed. Heib. {Vors. i 2 , p. 15.47). Simplicius 
adds : ' Now the sizes and distances of the sun and moon as determined up to 
now were ascertained (by calculations) starting from (observations of) eclipses, 
and the discovery of these things might reasonably be supposed to go back as 
far as Anaximander.' If by ' these things ' Simplicius means the use of the 
phenomena of eclipses for the purpose of calculating the sizes and distances of 
the sun and moon, his suggestion is clearly inadmissible. On Anaximander's 
theory eclipses of the sun and moon were caused by the stopping-up of the vents 
in their respective wheels through which the fire shone out ; moreover, the moon 
was itself bright and was not an opaque body receiving its light from the sun, 
notwithstanding the statement of Diogenes Laertius (ii. 1 ; Vors. i 2 , pp. 11. 40- 
12. 1) to the contrary; it is clear, therefore, that Anaximander's estimates of 
sizes and distances rested on no such basis as the observation of eclipses 
afforded to later astronomers. 

2 Diels, ' Uber Anaximanders Kosmos' in Archiv fur Gesch. d. Philosofihie, 
x, 1897, p. 231 ; cf. Tannery, p. 91. 



38 ANAXIMANDER part I 

than the sun and moon are, it is perhaps a fair inference that he 
would assume for a third hoop or ring containing stars an inner 
circumference representing 9 times the diameter of the earth ; the 
three rings would then have inner circumferences of 9, 18, 27, 
being multiples of 9 in arithmetical progression, while 9 is the 
square of 3 ; this is appropriate also to the proportion of 1 : 3 
between the depth of the disc representing the earth and the 
diameter of one of its faces. These figures suggest that they were 
not arrived at by any calculation based on geometrical construc- 
tions, but that we have merely an illustration of the ancient cult of 
the sacred numbers 3 and 9. 1 3 is the sacred number in Homer, 
9 in Theognis, 9 being the second power of 3. The cult of 3 and 
its multiples 9 and 27 is found among the Aryans, then among 
the Finns and Tartars, and next among the Etruscans (the Semites 
connected similar ideas with 6 and 7). Therefore Anaximander's 
figures really say little more than what the Indians tell us, namely 
that three Vishnu-steps reach from earth to heaven. 

The story that Anaximander was the first to discover the 
gnomon 2, (or sun-dial with a vertical needle) is incorrect, for 
Herodotos says that the Greeks learnt the use of the gnomon 
and the polos from the Babylonians. 3 Anaximander may, however, 
have been the first to ' introduce ' 4 or make known the gnomon in 
Greece, and to show on it ' the solstices, the times, the seasons, and 
the equinox'. 5 He is said to have set it up in Sparta. 6 He is 
also credited with constructing a sphere to represent the heavens, 7 
as was Thales before him. 8 

But Anaximander has yet another claim to undying fame. He 
was the first who ventured to draw a map of the inhabited earth. 9 
The Egyptians had drawn maps before, but only of particular dis- 
tricts ; 10 Anaximander boldly planned out the whole world with 
'the circumference of the earth and of the sea'. 11 Hecataeus, a 
much-travelled man, is said to have corrected Anaximander's map, 

1 Diels, loc. cit., p. 233. 2 Diog. L. ii. I (Vors. i 2 , p. 12. 3). 

3 Herodotus, ii. 109. * άσήγαγβ, Suidas (Vors. i 2 , p. 12. 18). 

5 Euseb. Praep. Evang. x. 14. II (Vors. i 2 , p. 12. 24). 

6 Diog. L. ii. 1. 7 Ibid. ii. 2. 8 Cic. De rep. i. 14. 22. 

9 Agathemerus (from Eratosthenes), i. 1 (Vors. i 2 , p. 12. 36). 

10 Gomperz, Griechische Denker, i 3 , pp. 41, 422. 

11 Diog. L. ii. 2 (Vors. i 2 , p. 12. 5). 



CH.iv ANAXIMANDER 39 

so that it became the object of general admiration. According to 
another account, Hecataeus left a written description of the world 
based on the map. In the preparation of the map Anaximander 
would of course take account of all the information which reached 
his Ionian home as the result of the many journeys by land and 
sea undertaken from that starting-point, journeys which extended 
to the limits of the then-known world ; the work involved of course 
an attempt to estimate the dimensions of the earth. We have, 
however, no information as to his results. 1 

Anaximander's remarkable theory of evolution does not concern ν 
us here. 2 

1 On Anaximander's map see Berger, Geschichte der wissenschaftlichen 
Erdkunde der Griechen, 2 ed., 1903, pp. 35 sqq. 

2 See Plut. Symp. viii. 8. 4 ( Vors. i 2 , p. 17. 24) ; Aet. v. 19. 4 (D. G. p. 430 ; Vors. 
i 2 , p. 17. 18) ; Ps. Plut. Stromal. 2 {D. G. p. 579) ; Hippol. Refut. i. 6. 6 {D. G. 

p. 560). According to Anaximander, animals first arose from slime evaporated αΛλ^Ο******* 
by the sun ; they first lived in the sea and had prickly coverings ; men 
at first resembled fishes. 




r 






ν 

ANAXIMENES 

For Anaximenes of Miletus (whose date Diels fixes at 585/4- 
528/4 B. C.) the earth is still flat, like a table, 1 but, instead of resting 
on nothing, as with Anaximander, it is supported by air, riding 
upon it, as it were. 2 Aristotle explains this assumption thus : 3 

' Anaximenes, Anaxagoras, and Democritus say that its flatness 
is what makes it remain at rest ; for it does not cut the air below 
it but acts like a lid to it, and this appears to be characteristic of 
those bodies which possess breadth. Such bodies are, as we know, 
not easily displaced by winds, because of the resistance they offer. 
The philosophers in question assert that the earth resists the air 
below it, in the same way, by its breadth, and that the air, on the 
other hand, not having sufficient space to move from its position, 
remains in one mass with that which is below it, just as the water 
does in water-clocks.' 

The sun, moon, and stars are evolved originally from earth ; for 
it is from earth that moisture arises ; then, when this is rarefied, fire 
is produced, and the stars are composed of fire which has risen 
aloft. 4 The sun, moon, and stars are all made of fire, and they ride 
on the air because of their breadth. 5 The sun is flat like a leaf; 6 
it derives its very adequate heat from its rapid motion. 7 The stars, 
on the other hand, fail to warm because of their distance. 8 

The stars are fastened on a crystal sphere, like nails or studs. 9 

1 Aet. iii. 10. 3 (D. G. p. 377 ; Vors. i 2 , p. 20. 26). 

2 Ps. Plut. Stromat. 3 (D. G. p. 580; Vors. i 2 , p. 18. 27); Hippol. Refut. i. 7. 
4 (D. G. p. 560; Vors. i 2 , p. 18. 40) ; Aet. iii. 15. 8 (D. G. p. 380; Vors. i 2 , 
p. 20. 34). 

3 De caelo ii. 13, 294b 13 (Vors. i 2 , p. 20. 27). 

4 Ps. Plut. Stromat. 3 (D. G. p. 580; Vors. i 2 , p. 18. 27) ; Hippol. Refut. i. 
7. 5 (D. G. p. 561 ; Vors. i 2 , p. 18. 42). 

5 Hippol., loc. cit. (Vors. i 2 , p. 18. 41). 

6 Aet. ii. 22. 1 (D. G. p. 352 ; Vors. i 2 , p. 20. 5). 

7 Ps. Plut. Stromat. 3 (D. G. p. 580; Vors. i 2 , p. 18. 28). 

8 Hippol., loc. cit. (Vors. i 2 , p. 19. 1). 

9 Aet. ii. 14. 3 (D. G. p. 344 ; Vors. i 2 , p. 19. 38). 



ANAXIMENES 41 

The stars do not move or revolve under the earth as some suppose, 
but round the earth, just as a cap can be turned round the head. 
The sun is hidden from sight, not because it is under the earth, but 
because it is covered by the higher parts of the earth and because 
its distance from us is greater. 1 With this statement may be com- 
pared the remark of Aristotle that 

1 Many of the ancient meteorologists were persuaded that the sun 
is not carried under the earth, but round the earth, and in particular 
our northern portion of it, and that it disappears and produces night 
because the earth is lofty towards the north.' 2 

The allusion is also to Anaximenes when we are told that some 
(i. e. Anaximenes) make the universe revolve like a millstone 
(μνλοζιδως), others (i.e. Anaximander) like a wheel. 3 

Now it is difficult to understand how the stars which, being fixed 
on a crystal sphere, move bodily with it round a diameter of the 
sphere, and which are seen to describe circles cutting the plane of 
the horizon at an angle, can do otherwise than describe the portion 
of their paths between their setting and rising again by passing 
under the earth ; and all sorts of attempts have been made to 
explain the contradiction. Schaubach pointed out that the circles 
described by the stars could not all converge and meet, say, on the 
horizon to the north ; for then they could not be parallel. 4 
Ottinger 5 supposed that the attachment of the stars to the crystal 
sphere only held good while they were above the horizon ; then, 
when they reached the horizon, they became detached and passed 
round in the plane of the horizon till they reached the east again ! 
Zeller, Martin, and Teichmiiller all have explanations which are 
more or less violent attempts to make ' under ' mean not exactly 
' under ', but something else. Teichmuller, 6 to explain the simile of 
the cap, observes that the ancients wore their caps, not as we wear 
our hats, but tilted back on the neck. The simile of the cap worn 

1 Hippol., loc. cit. (Vors. i 2 , pp. 18. 45-19. 1) ; cf. Aet. ii. 16. 6 {D. G. p. 346; 
Vors. i 2 , p. 19. 39). 5l 

2 Aristotle, Meteorologica ii. 1, 354 a 28. 

3 Aet. ii. 2. 4 {D. G. p. 329 b, note ; Vors. i 2 , p. 19. 32). 

4 Schaubach, Geschichte der griechischen Astronomie bis auf Eratosthenes, 

p. 136. 

5 Quoted by Sartorius, op. cit., p. 33. 

6 Teichmuller, Studien zur Geschichte der Begriffe, 1 874, p. 100. 



4 3 ANAXIMENES parti 

in this way would no doubt be appropriate if Anaximenes had 
confined his comparison to some stars only, namely those in the 
north which are always above the horizon and never set ; but he 
does not make this limitation ; and this view of the cap does not 
correspond very well to the revolution ' like a millstone '. 

More important is the distinction between the motion of the 
fixed stars, which are fastened like nails on the sphere, and 
the motion of the sun and moon. Anaximenes says that 

' The sun and the moon and the other stars float on the air on 
account of their breadth.' * 

This is intelligible as regards the sun, because it is like a leaf ; but, 
as regards ' the other stars ' it seems clear that floating on the air is 
inconsistent with their being fastened to the heavenly sphere ; it is 
almost necessary therefore to suppose that ' the other stars ' are 
here, not the fixed stars, but the planets, and that this ' floating on 
the air' is a hypothesis to explain the disagreement between the 
observed motions of the sun, moon, and planets on the one hand, 
and the simple rotation of the stars in circles on the other. We are 
told in another place that, while Anaximenes said that the stars are 
fastened like nails on the crystal sphere, c some ' say that they are 
'leaves of fire, like pictures'; 2 it is tempting, therefore, to read, 
instead of tvioi in the nominative, the accusative kviovs (αστέρα?), 
when the meaning would be ' but that some of the stars are leaves 
of fire', &c. The idea that the planets are meant in the above 
passage is further supported by another statement that 

1 The stars execute their turnings (τάς τροπας ποπΐσθαι) in conse- 
quence of their being driven out of their course by condensed air 
which resists their free motion.' 3 

It seems clear that the c turnings ' here referred to are not the 
1 solstices ', but simply the turnings of the stars in the sense of their 
revolution in their respective orbits, so far as they are not fixed on 
the crystal sphere ; 4 that is to say, the statement refers to the 
planets only. 



1 Hippol. Refut. i. 7. 4 (D. G. p. 561 ; Vors. i 2 , p. 18. 41). 

2 Aet. ii. 14. 4 (D. G. p. 344 ; Vors. i a , p. 19. 38). 

3 Aet. ii. 23. 1 (D. G. p. 352 ; Vors. i 2 , p. 20. 5). 

4 Zeller, i 5 , p. 250. 



CH.v ANAXIMENES 43 

It would seem certain therefore that Anaximenes was the first 
to distinguish the planets from the fixed stars in respect of their 
irregular movements, which he accounted for in the same way as 
the motions of the sun and moon. This being so, it seems not 
impossible that the passages about the sun and the stars not 
moving under , but laterally round, the earth refer exclusively to 
the sun, moon, and planets ; * the fact of their floating on the air 
might be supposed to be a reason why they should not ever fall 
below the earth, which itself rests on the air, and in this way the 
difficulty with regard to the motion of the fixed stars would 
disappear. 

Another improvement on the system of Anaximander is the 
relegation of the stars to a more distant region than that in which 
the sun moves. Anaximander had made the sun's wheel the most 
remote, the moon's next to it, and those of the stars nearer still 
to the earth ; Anaximenes, however, explains that the stars do not 
give warmth because they are too far off, and with this may be 
compared his statement that 

1 The rotation which is the furthest away from the earth is (that 
of) the heaven/ 2 

which view is attributed to him in common with Parmenides. 

Anaximenes made yet another innovation of some significance. 
He said that 

1 There are also, in the region occupied by the stars, bodies of an 
earthy nature which are carried round along with them,' 3 

and that, 

1 While the stars are of a fiery nature, they also include (or 
contain) certain earthy bodies which are carried round along with 
them but are not visible.' 4 

Zeller 5 interprets these passages as ascribing an earthy nucleus 
to the stars ; and this is not unnaturally suggested by the second 
of the two passages. But the first passage suggests another possible 

1 This was the suggestion of Heeren (Stobaeus, i, p. 511). 

2 Aet. ii. 11. 1 (D. G. p. 339 ; Vors. i 2 , p. 19. 34). 

8 Hippol. Refut. i. 7. 5 (D. G. p. 561 ; Vors. i 8 , p. 18. 44). 
4 Aet. ii. 13. 10 (D. G. p. 342; Vors. i a , p. 19. 36). 
6 Zeller, i 6 , pp. 247, 248. 



44 



ANAXIMENES parti 



interpretation ; bodies of an earthy nature in the region occupied 
by the stars (kv τω τόττω των αστέρων) might be separate from 
them and not 'contained in them', although carried round with 
them. ' The stars ' in the two passages no doubt include the sun 
and moon; but the sun is flat like a leaf; why then should 
Anaximenes attach to it an earthy substance as well ? The object 
of the invisible bodies of an earthy nature carried round along with 
' the stars ' is clearly to explain eclipses and the phases of the moon. 
If, then, Anaximenes supposed that one side in both the sun and 
the moon was bright and the other dark, his idea would doubtless 
be that they might sometimes turn their dark side to us in such 
a way as to hide from us more or less the bright side. (This was 
the idea of Heraclitus, though with him the heavenly bodies had 
not a flat surface but were hollowed out like a basin or bowl.) But 
the phenomena of eclipses are more simply accounted for if we 
suppose the earthy bodies of Anaximenes to be separate from the 
sun and moon, and to get in front of them ; we need not therefore 
hesitate to attribute to him this fruitful idea which ultimately led 
to the true explanation. Anaxagoras said that the moon is eclipsed 
because the earth is interposed, but, not being able to account for 
all the phenomena in this way, he conceived that eclipses were also 
sometimes due to obstruction by bodies ' below the moon ', which 
he describes in almost the same words as Anaximenes, namely as 
'certain bodies (in the region) below the stars which are carried 
round with the sun and moon and are invisible to us'. Clearly 
therefore Anaxagoras was indebted to Anaximenes for this con- 
ception ; and again the role of the counter-earth in the Pythagorean 
system is much the same as that of the 'earthy bodies' now in 
question. 

Tannery 1 goes further and maintains that Anaximenes' hypothesis 
was bound to lead to the true explanation of eclipses. ' For, if any 
one asked himself why these dark bodies were not seen at all, the 
question of their being illuminated by the sun would present itself, 
and it was easy to recognize that, under the most general conditions, 
the phenomena which such' a dark body would necessarily present 
were really similar to the phases of the moon. From this to the 

1 Tannery, Pour Vhistoire de la science hellene, pp. 153, 154. 



CH. v ANAXIMENES 45 

recognition of the fact that the moon itself is opaque there was only 
one step more. The role of the moon in regard to the eclipses 
of the sun was easy to deduce, while the question of the lighting 
up of the moon by the sun at night naturally brought into play the 
shadow of the earth and, through that, led to the discovery of the 
cause of eclipses of the moon. The hypothesis then of Anaximenes 
has a true scientific character, and constitutes for him a title to 
fame, the more rare because the conception appears to have been 
absolutely original, while his other ideas are not in general of the 
same stamp/ While the successive steps towards the discovery 
of the truth may no doubt have been taken in the order suggested, 
it must, I think, be admitted that, at the point where the question 
of the illumination of the opaque bodies by the sun would present 
itself (*se posait'), a very active imagination would be required to 
suggest the transition to this question ; and, even after the transition 
was made, it would be necessary to assume further that the opaque 
bodies are spherical in form, an assumption nowhere suggested by 
Anaximenes. 

Tannery * adds that the only feature of Anaximenes' system that 
was destined to an enduring triumph is the conception of the stars 
being fixed on a crystal sphere as in a rigid frame. Although 
attempts were made later to arrive at a more immaterial and less 
gross conception of the substance rigidly connecting the fixed stars, 
the character of this connexion was not modified, and the rigidity 
of the sphere really remained the fundamental postulate of all 
astronomy up to Copernicus. The exceptions to the general 
adoption of this view were, curiously enough, the Ionian physicists 
of the century immediately following Anaximenes. 

It would appear that Anaximenes anticipated the Pythagorean 
notion that the world breathes, for he says : 

* Just as our soul, being air, holds us together, so does breath and 
air encompass the whole world.' 2 

1 Tannery, op. cit., p. 154. 

2 Fragment in Aet. i. 3. 4 (£>. G. p. 278 ; Vors. i 2 , p. 21. 17). 



VI 

PYTHAGORAS 

Pythagoras, undoubtedly one of the greatest names in the 
history of science, was an Ionian, born at Samos about 
572 B.C., the son of Mnesarchus. He spent his early manhood 
in Samos, removed in about $$% B.C. to Croton, where he founded 
his school, and died at Metapontium at a great age (75 years 
according to one authority, 80 or more according to others). His 
interests were as various as those of Thales, but with the difference 
that, whereas Thales' knowledge was mostly of practical application, 
with Pythagoras the subjects of which he treats become sciences 
for the first time. Mathematicians know him of course, mostly 
or exclusively, as the reputed discoverer of the theorem of Euclid 
I. 47 ; but, while his share in the discovery of this proposition 
is much disputed, there is no doubt that he was the first to make 
theoretical geometry a subject forming part of a liberal education, 
and to investigate its first principles. 1 With him, too, began the 
Theory of Numbers. A mathematician then of brilliant achieve- 
ments, he was also the inventor of the science of acoustics, an 
astronomer of great originality, a theologian and moral reformer, 
founder of a brotherhood * which admits comparison with the orders 
of mediaeval chivalry/ 2 

The epoch-making discovery that musical tones depend on 
numerical proportions, the octave representing the proportion of 
2 : 1, the fifth 3 : 2, and the fourth 4 : 3, may with sufficient certainty 
be attributed to Pythagoras himself, 3 as may the first exposition 
of the theory of means, and of proportion in general applied to 
commensurable quantities, i.e. quantities the ratio between which 
can be expressed as a ratio between whole numbers. The all- 

1 Proclus, Comm. on Eucl. I, p. 65. 15-19. 

2 Gomperz, Griechische Denker, i 3 , pp. 80, 81. 

3 Burnet, Early Greek Philosophy, p. 118. 



PYTHAGORAS 47 

pervading character of number being thus shown, what wonder 
that the Pythagoreans came to declare that number is the essence 
of all things ? The connexion so discovered between number and 
music would also lead not unnaturally to the idea of the * harmony 
of the heavenly bodies \ 

Pythagoras left no written exposition of his doctrines, nor did 
any of his immediate successors in the school ; this statement is 
true even of Hippasus, about whom the different stones arose 
(i) that he was expelled from the school because he published 
doctrines of Pythagoras, 1 (2) that he was drowned at sea for 
revealing the construction of the dodecahedron in a sphere and 
claiming it as his own, 2 or (as others have it) for making known 
the discovery of the irrational or incommensurable. 3 Nor is the 
absence of any written record of early Pythagorean doctrine to 
be put down to any pledge of secrecy binding the school ; there 
does not seem to have been any secrecy observed at all unless 
perhaps in matters of religion or ritual ; the supposed secrecy 
seems to have been invented to explain the absence of any trace 
of documents before Philolaus. The fact appears to be merely 
that oral communication was the tradition of the school, and the 
closeness of their association enabled it to be followed without 
inconvenience, while of course their doctrine would be mainly too 
abstruse to be understood by the generality of people outside. 

Philolaus was the first Pythagorean to write an exposition of 
the Pythagorean system. He was a contemporary of Socrates and 
Democritus, probably older than either, and we know that he lived 
in Thebes in the last decades of the fifth century. 4 

It is difficult in these circumstances to disentangle the portions 
of the Pythagorean philosophy which may safely be attributed to 
the founder of the school. Aristotle evidently felt this difficulty ; 
he clearly knew nothing for certain of any ethical or physical 
doctrines going back to Pythagoras himself; and, when he speaks 
of the Pythagorean system, he always refers it to ' the Pythagoreans ', 
sometimes even to 'the so-called Pythagoreans'. 5 The account 

1 Clem. Stromat. v. 58 {Vors. i 2 , p. 30. 18); Iamblichus, Vit. Pyth. τ,φ, 247 
(Vors. i 2 , p. 30. 10, 14). 

2 Iamblichus, Vit. Pyth. 88 ( Vors. i 2 , p. 30. 2). 

3 Ibid. 247 (Vors. i 2 , p. 30. 17). 

4 Zeller, i 5 , pp. 337, 23^- 5 Burnet, Early Greek Philosophy, p. 100. 






4 8 PYTHAGORAS part ι 

which he gives of the Pythagorean planetary system corresponds 
to the system of Philolaus as we know it from the Doxographi. 

For Pythagoras's own system, therefore, that of Philolaus affords 
no guide ; we have to seek for traces, in the other writers of the 
end of the sixth and the beginning of the fifth centuries, of opinions 
borrowed from him or of polemics directed against him. 1 On these 
principles we have seen reason to believe that he was the first to 
maintain that the earth is spherical and, on the basis of this 
assumption, to distinguish the five zones. 

How Pythagoras came to conclude that the earth is spherical 
in shape is uncertain. There is at all events no evidence that he 
borrowed the theory from any non- Greek source. 2 On the assump- 
tion, then, that it was his own discovery, different suggestions 3 have 
been put forward as to the considerations by which Pythagoras 
convinced himself of its truth. One suggestion is that he may 
have based his opinion upon the correct interpretation of phenomena, 
and above all, on the round shadow cast by the earth in the eclipses 
of the moon. But it is certain that Anaxagoras was the first to 
suggest this, the true explanation of eclipses. The second possibility 
is that Pythagoras may have extended his assumption of a spherical 
sky to the separate luminaries of heaven ; the third is that his 
ground was purely mathematical, or mathematico-aesthetical, and 
that he attributed spherical shape to the earth for the simple reason 
that * the sphere is the most beautiful of solid figures '. 4 I prefer 
the third of these hypotheses, though the second and third have the 
point of contact that the beauty of the spherical shape may have 

1 Tannery, op. cit., p. 203. 

2 The question is discussed by Berger {Geschichte der wissenschaftlichen 
Erdkunde der Griechen, pp. 17 1-7) who is inclined to think that, along with the 
facts about the planets and their periods discovered, as the result of observations 
continued through long ages, by the Egyptians and Babylonians, the doctrine of 
a suspended spherical earth also reached the Greeks from Lydia, Egypt, or 
Cyprus. Berger admits, however, that Diodorus (ii. 31) denies to the Babylonians 
any knowledge of the earth's sphericity. Martin, it is true, in a paper quoted 
by Berger (p. 177, note), assumed that the Egyptians had grasped the idea of 
a spherical earth, but, as Gomperz observes {Griechische Denker, i 3 , p. 430), this 
assumption is inconsistent with the Egyptian representation of the earth's shape 
as explained by one of the highest authorities on the subject, Maspero, in his 
Hist, ancienne des peupies de Γ Orient classique, Les origines, pp. 16, 17. 

3 Gomperz, Griechische Denker^ i 3 , p. 90. 

4 Diog. L. viii. 35 (Vors. i 2 , p. 280. 1) attributes this statement to the 
Pythagoreans. 



ch. vi PYTHAGORAS 49 

dictated its application both to the universe and to the earth. But, 
whatever may have been the ground, the declaration that the earth 
is spherical was a great step towards the true, the Copernican 
view of the universe. 1 It may well be (though we are not told) 
that Pythagoras, for the same reason, gave the same spherical 
shape to the sun and moon and even to the stars, in which case 
the way lay open for the discovery of the true cause of eclipses and 
of the phases of the moon. 

There is no doubt that Pythagoras's own system was geocentric. 
The very fact that he is credited with distinguishing the zones is 
an indication of this ; the theory of the zones is incompatible with 
the notion of the earth moving in space as it does about the central 
fire of Philolaus. But we are also directly told that he regarded 
the universe as living, intelligent, spherical, enclosing the earth 
in the middle^ the earth, too, being spherical in shape. 2 Further, 
it seems clear that he held that the universe rotated about an axis 
passing through the centre of the earth. Thus we are told by 
Aristotle that 

* Some (of the Pythagoreans) say that time is the motion of the 
whole (universe), others that it is the sphere itself ' ; 3 

and by Aetius that 

1 Pythagoras held time to be the sphere of the enveloping 
(heaven).' 4 

Alcmaeon, a doctor of Croton, although expressly distinguished 
from the Pythagoreans by Aristotle, 5 is said to have been a pupil 
of Pythagoras ; 6 even Aristotle says that, in the matter of the 
Pythagorean pairs of opposites, Alcmaeon, who was a young man 
when Pythagoras was old, expressed views similar to those of the 
Pythagoreans, • whether he got them from the Pythagoreans or they 
from him'. 7 Hence he was clearly influenced by Pythagorean 

1 Gomperz, Griechische Denker, i 3 , p. 90. 

2 Alexander Polyhistor in Diog. L. viii. 

3 Aristotle, Phys. iv. 10, 218 a 33. 

4 Aet. i. 21. I {D. G. p. 318 ; Vors. i 2 , p. 277. 19). 

5 Aristotle, Metaph. A. 5, 986 a 27-31. 

6 Diog. L. viii. 83 (Vors. i 2 , p. loo. 19) ; Iamblichus, Vit. Pyth. 104. 

7 Aristotle, Metaph. i. 5, 986 a 28. 

1410 Ε 



5o PYTHAGORAS part ι 

doctrines. Now the doxographers' account of his astronomy includes 
one important statement, namely that 

1 Aicmaeon and the mathematicians hold that the planets have 
a motion from west to east, in a direction opposite to that of the 
fixed stars.' * 

Incidentally, the assumption of the motion of the fixed stars 
suggests the immobility of the earth. But this passage is also the 
first we hear of the important distinction between the diurnal 
revolution of the fixed stars from east to west and the independent 
movement of the planets in the opposite direction ; the Ionians say 
nothing of it (though perhaps Anaximenes distinguished the planets 
as having a different movement from that of the fixed stars) ; 
Anaxagoras and Democritus did not admit it; the discovery, 
therefore, appears to belong to the Pythagorean school and, in view 
of its character, it is much more likely to have been made by the 
Master himself than by the physician of Croton. 2 For the rest 
of Alcmaeon's astronomy is on a much lower level ; he thought 
the sun was flat, 3 and, like Heraclitus, he explained eclipses and 
the phases of the moon as being due to the turning of the moon's 
bowl-shaped envelope. 4 It is right to add that Burnet 5 thinks 
that the fact of the discovery in question being attributed to 
Aicmaeon implies that it was not due to Pythagoras. Presumably 
this is inferred from the words of Aristotle distinguishing Aicmaeon 
from the Pythagoreans ; but either inference is possible, and 
I prefer Tannery's. It is difficult to account for Aicmaeon being 
credited with the discovery if, as Burnet thinks, it was really Plato's. 

But we have also the evidence of Theon of Smyrna, who states 
categorically that Pythagoras was the first to notice that the 
planets move in independent circles : 

' The impression of variation in the movement of the planets 
is produced by the fact that they appear to us to be carried through 
the signs of the zodiac in certain circles of their own, being fastened 
in spheres of their own and moved by their motion, as Pythagoras 

1 Aet. ii. 16. 2-3 (D. G. p. 345 ; Vors. i 2 , p. 101. 8). 

2 Tannery, op. cit., p. 208. 

3 Aet. ii. 22. 4 (D. G. p. 352 ; Vors. i 2 , p. 101. 10). 

4 Aet. ii. 29. 3 (D. G. p. 359; Vors. i 2 , p. 101. 10-12). 
6 Burnet, Early Greek Philosophy, p. 1 23, note. 



ch.vi PYTHAGORAS 51 

was the first to observe, a certain varied and irregular motion being 
thus grafted, as a qualification, upon their simply and uniformly 
ordered motion in one and the same sense ' [i. e. that of the daily 
rotation from east to west]. 1 

It appears probable, therefore, that the theory of Pythagoras 
himself was that the universe, the earth, and the other heavenly 
bodies are spherical in shape, that the earth is at rest in the centre, 
that the sphere of the fixed stars has a daily rotation from east to 
west about an axis passing through the centre of the earth, and 
that the planets have an independent movement of their own in 
a sense opposite to that of the daily rotation, i.e. from west to east. 

1 Theon of Smyrna, p. 150. 12-18. 



Ε 2 



VII 

XENOPHANES 

Xenophanes of Colophon was probably born about 570 and 
died after 478 B.C. What we know for certain is that he spoke of 
Pythagoras in the past tense, 1 that Heraclitus mentions him along 
with Pythagoras, 2 and that he says of himself that, from the time 
when he was 25 years of age, three-score years and seven had 
1 tossed his care-worn soul up and down the land of Hellas.' 3 
He may have left his home at the time when Ionia became a Persian 
province (545 B.C.) and gone with the Phocaeans to Elea, 4 founded 
by them in 540/39 B. C, six years after they left Phocaea. 5 As he was 
writing poetry at 92 and is said to have been over 100 when he 
died, 6 the above dates are consistent with the statement that he was 
a contemporary of Hieron, who reigned from 478 to 467 B.C. 7 
According to Theophrastus, he had 'heard* Anaximander. 8 

Xenophanes was more a poet and satirist than a natural philo- 
sopher, but Heraclitus credited him with wide learning, 9 and he 
is said to have opposed certain doctrines of Pythagoras and Thales. 10 
We are told that he wrote epics as well as elegies and iambics 
attacking Homer and Hesiod. In particular, 2,000 verses on the 
foundation of Colophon and the settlement at Elea are attributed 
to him. 11 He is supposed to have written a philosophical poem ; 
Diels refers about sixteen fragments to such a poem, to which the 

1 Fr. 7 ( Vors. i 2 , p. 47. 20-23). 2 Heraclitus, Fr. 40 ( Vors. i 2 , p. 68. 10). 

3 Fr. %(Vors.\\ p. 48.3-6). 

4 Gomperz, Griechische Denker, i 3 , pp. 127, 436. 

5 Herodotus, i. 164-7. 

6 Censorinus, De die natali c. 15. 3, p. 28. 21, ed. Hultsch. 

7 Timaeus in Clem. Stromat. i. 14, p. 353 ( Vors. i 2 , p. 35. 2). 

8 Diog. L. ix. 21 (Vors. i 2 , p. 34. 35). 

9 Heraclitus, loc. cit.: 'Wide learning does not teach one to have under- 
standing ; if it did, it would have taught Hesiod and Pythagoras, and again 
Xenophanes and Hecataeus.' 

Diog. L. ix. 18 (Vors. i 2 , p. 34. 12). n Ibid. ix. 20 (Vors. i 2 , p. 34. 26). 



in 



XENOPHANES 53 

name On Nature (Περί φύσζως) was given ; but such titles are 
of later date than Xenophanes, and Burnet 1 holds that all the 
fragments might have come into the poems directed against Homer 
and Hesiod, the fact that a considerable number of them come 
from commentaries on Homer being significant in this connexion. 

Xenophanes attacked the popular mythology, proving that God 
must be one, not many (for God is supreme and there can only 
be one supreme power), 2 eternal and not born (for it is as impious 
to say that the gods are born as it would be to say that they die ; 
in either case there would be a time when the gods would not be\ ; 3 
he reprobated the scandalous stories about the gods in Homer and 
Hesiod 4 and ridiculed the anthropomorphic view which gives the 
gods bodies, voices, and dress like ours, observing that the Thracians 
made them blue-eyed and red-haired, the Aethiopians snub-nosed 
and black, 5 while, if oxen or horses or lions had hands and could 
draw, they would draw them as oxen, horses, and lions respectively. 6 
God is the One and the All, the universe ; 7 God remains unmoved 
in one and the same place ; 7 God is eternal, one, alike every way7 
finite, spherical and sensitive in all parts, 9 but does not breathe. 10 
It is difficult to reconcile the finite and spherical God with 
Xenophanes' description of the world, which may be summarized 
as follows. 

The world was evolved from a mixture of earth and water, 11 
and the earth will gradually be dissolved again by moisture ; this 
he infers from the fact that shells are found far inland and on 
mountains, and in the quarries of Syracuse there have been found 
imprints (fossils) of a fish and of seaweed, 12 and so on, these 
imprints showing that everything was covered in mud long ago, 

1 Burnet, Early Greek Philosophy, p. 128. 

2 Simpl. in Phys. p. 22. 31 (Vors. i 2 , p. 40. 30). 

3 Aristotle, Rhetoric ii. 23, 1399 b 6. 

1 Fr. 11 (Vors. i 2 , p. 48. 13). ' 5 Fr. 14, 16 [Vors. i 2 , p. 49. 2, 11). 

β Fr. 15 ( Vors. i 2 , p. 49. 5). 

7 Aristotle, Metaph. A. 5, 986 b 21 (Vors. i 2 , p. 40. 15) ; Simpl., loc. cit. (Vors. 
i 2 , p. 40. 29) ; cf. Cicero, De nat. deor. i. 1 1. 28 ( Vors. i 2 , p. 41. 44) ; Acad.pr. ii. 37. 
118 (Vors. i 2 , p. 41. 42). 

8 Fr. 26 (Vors. i 2 , p. 50. 22). 

9 Hippol. Refut. i. 14. 2 (D. G. p. 565 ; Vors. i 2 , p. 41. 26). 

10 Diog. L. ix. 19 (Vors. i 2 , p. 34. 18). 

11 Fr. 29. 33 (Vors, i 2 , p. 51. 5, 20). 

12 I read, with Burnet, after Gomperz φνκωρ (seaweed) instead of φωκων. 



54 XENOPHANES parti 

and that the imprints dried on the mud. All men will disappear 
when the earth is absorbed into the sea and becomes mud, after 
which the process of coming into being starts again ; all the worlds 
(alike) suffer this change. 1 This is, of course, the theory of 
Anaximander. 

As regards the earth we are told that 

' This upper side of the earth is seen, at our feet, to touch the air, 
but the lower side reaches to infinity.' 2 

'This is why some say that the lower portion of the earth is 
infinite, asserting, as Xenophanes of Colophon does, that its roots 
extend without limit, in order that they may not have the trouble 
of investigating the cause (of its being at rest). Hence Empedocles' 
rebuke in the words " if the depths of the earth are without limit 
and the vast aether (above it) is so also, as has been said by the 
tongues of many and vainly spouted forth from the mouths of men 
who have seen little of the whole ".' 3 

c Xenophanes said that on its lower side the earth has roots 
extending without limit.' 4 

1 The earth is infinite, and is neither surrounded by air nor by the 
heaven.' 5 

Simplicius 6 (on the second of the above passages) observes that, 
not having seen Xenophanes' own verses on the subject, he cannot 
say whether Xenophanes meant that the under side of the earth 
extends without limit, and that this is the reason why it is at rest, or 
meant to assert that the space below the earth, and the aether, is 
infinite, and consequently the earth, though it is in fact being carried 
downwards without limit, appears to be at rest ; for neither Aristotle 
nor Empedocles made this clear. Presumably, however, as 
Simplicius had not seen Xenophanes' original poem, he had not 
seen Fr. 28, the first of the above passages ; for this passage seems 
to be decisive ; there is nothing in it to suggest motion downwards, 
and, if it meant that there was infinite air below the earth as there 
is above, there would be no contrast between the upper and the 
under side such as it is the obvious intention of the author to draw. 7 

1 Hippol. Refut. i. 14. 5-6 (D. G. p. 566 ; Vors. i 2 , p. 41. 33-41). 

2 Fr. 28 {Vors. i 2 , p. 51. 2). 

s Aristotle, De caelo ii. 13, 294 a 21-28. 

4 Aet. iii. 9. 4 ; 11. 1,2 (D.G. pp. 376, 377 ; Vors. i 2 , p. 43. 33, 35). 

5 Hippol. Refut. i. 14. 3 (D. G. p. 565 ; Vors. i 2 , p. 41. 29). 

6 Simplicius on De caelo, p. 522. 7, ed. Heib. ( Vors. i 2 , p. 43. 28). 

7 As witness the μ*ν and the 84 and the clear opposition of ' touching the air ' 



CH. vii XENOPHANES 55 

According to Xenophanes the stars, including comets and 
meteors, are made of clouds set on fire; they are extinguished 
each day and are kindled at night like coals, and these happenings 
constitute their setting and rising respectively. 1 The so-called 
Dioscuri are small clouds which emit light in virtue of the motion, 
whatever it is, that they have. 2 

Similarly the sun is made of clouds set on fire ; clouds formed 
from moist exhalation take fire, and the sun is formed from the 
resulting fiery particles collected together. 3 The moon is likewise 
so formed, the cloud being here described as * compressed ' 
{πιπιλημένου) , 4 following an expression of Anaximander's for 
compressed portions of air ; the moon's light is its own. 5 

When the sun sets, it is extinguished, and when it next rises, it is 
a fresh one ; it is likewise extinguished when there is an eclipse. 6 

(ψ pi προσπλάζον) and reaching to infinity {is άπειρον Ικνύται). ικνύται implies, 
not motion towards, but arrival at, or reaching, a destination. Berger, it is true, 
rejects as a misapprehension the whole of the traditional view of Xenophanes' 
system {Gesch.der wissenschaftlichen Erdkunde der Griechen, pp. 191 sqq.). We 
shall have to consider his argument later ; but it necessitates getting a sense out 
of the fragment and the passage of Aristotle other than the literal interpretation. 
The significant words in the passage of Aristotle are ' saying that it (the earth) 
is rooted ad infinitum (eV άπειρον ερριζωσθαι) '. Berger (p. 194, note) holds that the 
expression is not used in the literal sense of having roots extending adinfinilum> 
but that ' we use the word έρριζώσθαι only as an expression for a supporting force 
not capable of closer definition ' ; he can only quote in favour of this certain 
metaphorical uses of ρίζα * root ' and other words connected with it, ριζώματα and 
ριζώδες, which of course do not in the least prove that έρριζώσθαι is used in 
a metaphorical sense in our passage ; indeed, if it is used in so vague a sense, it 
is difficult to see how Xenophanes thereby absolved himself from giving a further 
explanation of the cause of the earth's remaining at rest, which, according to 
Aristotle, was his object. As regards the fragment from Xenophanes' own 
poem, Berger says that he prefers to regard it as an attempt to give in few 
words an idea of the horizon which divides earth and heaven into an upper, 
visible, half, and an invisible lower half. This again leaves no contrast between 
the upper and lower sides of the earth such as the fragment is obviously intended 
to draw. On both points Berger's arguments are of the nature of special 
pleading, which can hardly carry conviction. 

1 Aet. ii. 13. 14, "i. 2. II {D. G. pp. 343, 367 ; Vors. i 2 , pp. 42. 39, 43. 15). 

2 Aet. ii. 18. ι (Ζλ c7. p. 347; Vors. i 2 , p. 42. 42). 

3 Aet. ii. 20. 3 {D. G. p. 348 ; Vors. i 2 , p. 42. 45) ; Hippol. Refut. i. 14. 3 {D. G. 
p. 565). 

4 Aet. ii. 25. 4 {D. G. p. 356 ; Vors. i 2 , p. 43. 12). 

5 Aet. ii. 28. I {D. G. p. 358 ; Vors. i 2 , p. 43. 13). 

6 Aet. ii. 24. 4 {D. G. p. 354 ; Vors. i 2 , p. 43. 1). The passage, which is under 
the heading • On eclipse of the sun ', implies that it is an eclipse which comes 
about by way of extinguishment {κατά σβέσιν), but the next words to the effect 
that the sun is a new one on rising again suggest that it is 'setting' rather 
than ' eclipse ', which should be understood. 



$6 XENOPHANES parti 

The phases of the moon are similarly caused by (partial) ex- 
tinction. 1 

According to Xenophanes, the sun is useful with reference to the 
coming into being and the ordering of the earth and of living things 
in it ; the moon is, in this respect, otiose. 2 

More remarkable are Xenophanes' theory of a multiplicity of suns 
and moons, and his view of the nature of the sun's motion ; and 
here it is necessary to quote the actual words of Aetius : 

' Xenophanes says that there are many suns and moons according 
to the regions (κλίματα), divisions (άποτομαί) and zones of the 
earth ; and at certain times the disc lights upon some division of 
the earth not inhabited by us and so, as it were, stepping on 
emptiness, suffers eclipse. 

1 The same philosopher maintains that the sun goes forward ad 
infinitum,^ and that it only appears to revolve in a circle owing to its 
distance (away from us).' 3 

The idea that the sun, on arriving at an uninhabited part of the 
earth, straightway goes out, as it were, is a curious illustration of 
the final cause. 4 For the rest, the passage, according to the most 
natural interpretation of it, implies that the sun does not revolve 
about the earth in a circle, but moves in a straight line ad infinitum, 
that the earth is flat, and that its surface extends without limit. 
On this interpretation we are presumably to suppose that the sun 
of any one day passes out of our sight and is seen successively in 
regions further and further distant towards the west until it is finally 
extinguished, while in the meantime the new sun of the next day 
follows the first, at an interval of 24 hours, over our part of the 
earth, and so on, with the result that at any given time there are 
many suns all travelling in the same straight direction ad infinitum. 
If this is the correct interpretation of Xenophanes' theory (and this 
is the way in which it is generally understood), it shows no advance 
upon, but a distinct falling off from, the systems of Anaximander 
and Anaximenes. Berger, 5 deeming it incredible that Xenophanes 
could have put forward views so crude, not to say childish, at 
a time when the notion of the sphericity of the earth discovered by 

1 Aet. ii. 29. 5 (D. G. p. 360 ; Vors. i 2 , p. 43. 14). 

2 Aet. ii. 30. 8 (D. G. p. 362 ; Vors. i 2 , p. 43. 9). 

8 Aet. ii. 24. 9 (D. G. p. 355 ; Vors. i 2 , p. 43. 3-8). 

4 Tannery, op. cit., p. 133. 5 Berger, op. cit., pp. 190 sqq. 



ch. νπ XENOPHANES 57 

the earliest Pythagoreans and by Parmenides must already have 
spread far and wide, seeks to place a new interpretation upon the 
passages in question. 

For the Ionians, with their flat earth, there was necessarily one 
horizon, so that the solar illumination and the length of the day 
were the same for all parts of the inhabited earth. As soon, how- 
ever, as the spherical shape of the earth was realized, it would 
necessarily appear that there were different horizons according to 
the particular spot occupied by an observer on the earth's surface. 
It was then, argues Berger, the different horizons which Xenophanes 
had in view when he spoke of many suns and moons according to 
the different regions or climates, divisions and zones of the earth ; 
he realized the difference in the appearances and the effects of the 
same phenomena at different places on the earth's surface, and he 
may have been the first to introduce, in this way, the mode of 
expression by which we commonly speak of different suns, the 
tropical sun, the Indian sun, the midnight sun, and the like. This 
is ingenious, but surely not reconcilable with other elementary 
opinions stated by Xenophanes, such as that there is a new sun 
every day. Then again, Berger has to explain the sun's 'going- 
forward ad infinitum ' as contrasted with circular motion ; as, on 
his theory, it cannot be motion in a straight line without limit, he 
takes it to be the motion in a spiral which the sun actually exhibits 
owing to the combination of its two motions, that of the daily 
rotation, and its yearly motion in the ecliptic, which causes a slight 
change in its latitude day by day. But in the first place this 
motion in a spiral is not motion forward ad infinitum, for the spiral 
returns on itself in a year just as a simple circular motion would in 
24 hours. Indeed, Berger's interpretation would make Xeno- 
phanes' system purely Pythagorean, and advanced at that, for 
we do not hear of the spiral till we find it in Plato. 1 And, if 
Heraclitus's system also represents (as we shall find it does) a set- 
back in astronomical theory, why should not Xenophanes' ideas 
have been equally retrograde ? 

There remains the story that Xenophanes told of an eclipse of 
the sun which lasted a whole month. 2 Could he have intended, by 

1 Plato, Timaeus 39 A. 

2 Aet. ii. 24. 4 (Z>. G. p. 354 ; Vors. i 2 , p. 43. 2-3). 



j 58 XENOPHANES 

this statement, to poke fun at Thales? 1 Berger, full of his theory 
that Xenophanes' ideas were based on the sphericity of the earth, 
thinks that he must have inferred that the length of the day would 
vary in different latitudes and according to the position of the sun 
in the ecliptic, and must have seen that, at the winter solstice for 
example, there would be a point on the earth's surface at which the 
longest night would last 24 hours, another point nearer the north 
pole where there would be a night lasting a month, and so on, and 
finally that at the north pole itself there would be a night six 
months long as soon as the sun passes to the south of the equator ; 
Xenophanes therefore, according to Berger, must simply have been 
alluding to the existence of a place where a night may last a month. 
If, as seems certain, Xenophanes' earth was flat, this explanation 
too must fall to the ground. 

1 Tannery, op. cit., p. 132. 






VIII 

HERACLITUS 

If the astronomy of Xenophanes represents a decided set-back 
in comparison with the speculations of Anaximander and Anaxi- 
menes, this is still more the case with Heraclitus of Ephesus 
(fl. 504/0, and therefore born about 544/0 B.C.) ; he was indeed no 
astronomer, and he scarcely needs mention in a history of astronomy 
except as an illustration of the vicissitudes, the ups and downs, 
through which a science in its beginnings may have to pass. Hera- 
clitus's astronomy, if it can be called such, is of the crudest descrip- 
tion. He does not recognize daily rotation ; he leaves all the 
apparent motions of the heavenly bodies to be explained by a 
continued interchange of matter between the earth and the heaven. 1 
His original element, fire, condenses into water, and water into 
earth ; this is the downward course. The earth, on the other hand, 
may partly melt ; this produces water, and water again vaporizes 
into air and fire ; this is the upward course. There are two kinds 
of exhalations which arise from the earth and from the sea ; the one 
kind is bright and pure, the other dark ; night and day, the months, 
the seasons of the year, the years, the rains and the winds, &c, are 
all produced by the variations in the proportion between the two 
exhalations. In the heavens are certain basins or bowls (σκάφαι) 
turned with their concave sides towards us, which collect the bright 
exhalations or vaporizations, producing flames ; these are the 
stars. 2 The sun and the moon are bowl-shaped, like the stars, and 
they are similarly lit up. 3 The flame of the sun is brightest 
and hottest ; the other stars are further away from the earth and 

1 Tannery, op. cit., pp. 168, 169. 

2 Diog. L. ix. 9-10 ( Vors. i 2 , pp. 55. 25 sqq.) ; cf. Aet. ii. 28. 6 (D. G. p. 359 ; 
Vors. i 2 , p. 59. 8). 

3 Aet. ii. 22. 2 ; 27. 2 ; 28. 6 (Ζλ G. pp. 352, 358, 359; Vors. i 2 , p. 59. 4, 6, 
7-10). 



6o HERACLITUS parti 

consequently they give out less light and warmth. The moon, 
although nearer the earth, moves in less pure air and is conse- 
quently dimmer than the sun ; the sun itself moves in pure and 
transparent air and is at a moderate distance from us, so that it 
warms and illuminates more. 1 ' If there were no sun, it would be 
night for anything the other stars could do.' 2 Both the sun and 
the moon are eclipsed when the bowls are turned upwards (i.e. so 
that the concave side faces upwards and the convex side faces in 
our direction); the changes in the form of the moon during the 
months are caused by gradual turning of the bowl. 3 

According to Heraclitus there is a new sun every day, 4 by which 
is apparently meant that, on setting in the west, it is extinguished 
or spent, 5 and then, on the morrow, it is produced afresh in the 
east by exhalation from the sea. G 

The question arises, what happens to the bowl or basin supposed 
to contain the sun if the sun has to be re-created in this way each 
morning? Either a fresh envelope must be produced every day 
for the rising of the sun in the east or, if the envelope is supposed 
to be the same day after day, it must travel round from the west to 
the east, presumably in the encircling water, laterally. 7 Diogenes 
Laertius (i. e. in this case Theophrastus) complains that Heraclitus 

1 Diog. L., loc. cit. ; Aet. ii. 28. 6 {D. G. p. 358 ; Vors. i 2 , p. 59. 10). 

2 Plutarch, Defort. 3, p. 98 c ( Vors. i 2 , p. 76. 8). 

3 Diog. L., loc. cit. ; Aet. ii. 24. 3 (£>. G. p. 354 ; Vors. i 2 , p. 59. 5). The 
explanation that the hollow side of the basins is turned towards us itself shows 
how crude were the ideas of Heraclitus. For it is clear that to account for the 
actual variations which we see in the shape of the moon, it is the outer side of 
a hemispherical bowl which should be supposed bright and turned towards us 
when the moon is full. 

4 Aristotle, Meteor, ii. 2, 355 a 14. 

5 Plato, Rep. vi. 498 A. 

6 Aristotelian Problems^ xxiii. 30, 934 b 35. It is true that a certain passage 
of Aristotle may be held to imply that Heraclitus did not maintain that the 
moon and the stars, as well as the sun, are fed and renewed by exhalations. 
Aristotle {Meteor, ii. 2, 354 b 33 sqq.) is speaking of those who maintain that 
the sun is fed by moisture. He first argues that, although fire may be said to 
be nourished by water (the flame arising through continuous alternation between 
the moist and the dry), this cannot take place with the sun ; ' and if the sun 
were fed in this same way, then it is clear that not only is the sun new every 
day, as Heraclitus says, but it is continuously becoming new (every moment) ' 
(355 a 11-15). * And,' he adds (355 a 18-21), ' it is absurd that these thinkers 
should only concern themselves with the sun, and neglect the conservation of 
the other stars, seeing that their number and their size is so great.' 

7 Zeller, i 5 , p. 684. 



ch.viii HERACLITUS 61 

gave no information as to the nature of these cups or basins. The 
idea, however, of the sun and moon being carried round in these 
σκάφαι reminds us forcibly of the Egyptian notion of the sun in 
his barque floating over the waters above, accompanied by a host 
of secondary gods, the planets and the fixed stars. 1 

Heraclitus held (as Epicurus did long afterwards) that the 
diameter of the sun is one foot, 2 and that its actual size is the 
same as its apparent size. 3 This in itself shows that Heraclitus 
was no mathematician ; as Aristotle says, ' it is too childish to 
suppose that each of the moving heavenly bodies is small in size 
because it appears so to us observing it from where we stand.' 4 

He called the arctic circle by the more poetical name of 'the 
Bear ', saying that ' the Bear represents the limits of morning and 
evening ' . . . whereas of course it is the arctic circle, not the Bear 
itself, which is the confine of setting and rising 5 (i.e. the stars 
within the arctic circle never set). 

According to Diogenes Laertius, Heraclitus said absolutely 
nothing about the nature of the earth ; 6 but we may judge that 
in his conception of the universe he was closer to Thales than to 
Anaximander; that is, he would regard the universe as a hemi- 
sphere rather than a sphere, and the base of the hemisphere as 
a plane containing the surface of the earth surrounded by the 
sea ; if he recognized a subterranean region, under the name of 
Hades, he does not seem to have formed any idea with regard to 
it beyond what was contained in the current mythology. 7 

When he gave 10,800 solar years as the length of a Great Year, 8 
he meant no astronomical Great Year, but the period of duration 
of the world from its birth to its resolution again into fire and 
vice versa. He arrived at it, apparently, by taking a generation 
of 30 years as a day and multiplying it by 360 as the number of 
days in a year. 

1 See pp. 19, 20 above. 2 Aet. ii. 21. 4 (D. G. p. 351 ; Vors. i 2 , p. 62. 7). 

3 Diog. L. ix. 7 (Vors. i 2 , p. 55. 12). 

4 Aristotle, Meteor, i. 3, 339 b 34. 

5 Strabo, i. 1. 6, p. 3 (Vors. i 2 , p. 78. 15). 
R Diog. L. ix. 11 (Vors. i 2 , p. 55. 46). 

7 Tannery, op. cit., p. 169. 

8 Aet. ii. 32. 3 (D.G. p. 364; Vors. i 2 , p. 59. 13); Censorinus, De die natali 
1 8. 11 (Vors. i 2 , p. 59. 16). 

Tannery, op. cit., p. 168. 



IX 

PARMENIDES 

With regard to the date of Parmenides there is a conflict of 
authority. On the one hand Plato says that Parmenides and Zeno 
paid a visit to Athens, Parmenides being then about 65 and Zeno 
nearly 40 years of age, and that Socrates, who was then very 
young (σφόδρα veos), conversed with them on this occasion. 1 Now 
if we assume that Socrates was about 18 or 20 years of age at 
this time, the date of the meeting would be about 451 or 449 B.C., 
and this would give 516 or 514 as the date of Parmenides' birth. On 
the other hand, Diogenes Laertius 2 says (doubtless on the authority 
of Apollodorus) that Parmenides flourished in 01. 69 (504/0 B.C.), 
in which case he must have been born about 540 B.C. In view of 
the number of cases in which, for artistic reasons, Plato indulged in 
anachronisms, it is not unnatural to feel doubt as to whether the 
meeting of Socrates with Parmenides was a historical fact. Zeller 3 
firmly maintained that it was a poetic fiction on the part of Plato ; 
but Burnet, on grounds which seem to be convincing, accepts it 
as a fact, exposing at the same time the rough and ready methods 
on which Apollodorus proceeded in fixing his dates. 4 

1 Plato, Parmenides 127 A-c. 2 Diog. L. ix. 23 ( Vors. i 2 , p. 106. 10). 

3 Zeller, i 5 , pp. 555, 556. 

4 Burnet, Early Greek Philosophy, pp. 192, 193. The story was early questioned. 
Athenaeus (xi. 1 5, p. 505 F; Vors. i 2 , p. 106. 47) doubted whether the age of Socrates 
would make it possible for him to have conversed with Parmenides or at any rate 
to have held or listened to such a discourse. But Plato refers to the meeting in 
two other places (Theaet. 183 E, Sophist 217 c), and (as Brandis and Karsten 
also pointed out) we should have to assume a deliberate falsification of facts on 
the part of Plato if he had inserted these two allusions solely for the purpose of 
inducing people to believe a fiction contained in another dialogue. We have, 
too, independent evidence of the visit of Zeno to Athens. Plutarch {Pericles 
4. 3) says that Pericles ' heard ' Zeno. The date given by Apollodorus, on the 
other hand, seems to be based solely on that of the foundation of Elea. As he 
adopts that date as the floruit of Xenophanes, so he makes it the date of 
Parmenides' birth. In like manner he makes Zeno's birth contemporaneous 



PARMENIDES 63 

Parmenides is said to have been a disciple of Xenophanes ; * he 
was also closely connected with the Pythagorean school, being 
specially associated with a Pythagorean, Ameinias Diochaites, for 
whom he conceived such an affection that he erected a ήρωον to t 

him after his death ; 2 Proclus quotes Nicomachus as authority 
for the statement that he actually belonged to the school, 3 and 
Strabo has a notice to the same effect. 4 It is not therefore 
unnatural that Parmenides' philosophical system had points in 
common with that of Xenophanes, while his cosmogony was on 
Pythagorean lines, with of course some differences. Thus his 
Being corresponds to the One of Xenophanes and, like it, is a well- 
rounded sphere always at rest ; he excluded, however, any idea 
of its infinite extension ; according to Parmenides it is definitely 
limited, rounded off on all sides, extending equally in all directions 
from the centre. 5 Parmenides differs from Xenophanes in denying 
genesis and destruction altogether ; these phenomena, he holds, 
are only apparent. 6 Being is identified with Truth ; anything else 
is Not-Being, the subject of opinion. Physics belongs to the latter 
deceptive domain. 7 

The main difference between the cosmologies of Parmenides and 
the Pythagoreans appears to be this. It seems almost certain that 
Pythagoras himself conceived the universe to be a sphere, and 
attributed to it daily rotation round an axis 8 (though this was 
denied by Philolaus afterwards); this involved the assumption 
that it is itself finite but that something exists round it ; the 
Pythagoreans, therefore, were bound to hold that, beyond the 
finite rotating sphere, there was limitless void or empty space; 

with Parmenides' floruit, thereby making Zeno forty years younger than Par- 
menides, whereas Plato makes him about twenty-five years younger. Burnet 
quotes E. Meyer {Gesch. des Alterth. iv. § 509, note) in support of his view. 

1 Aristotle, Metaph. A. 5, 986b 22 ; Simplicius, In phys. p. 22. 27 ( Vors. i 2 , p. 107. 
16). Diog. L. (ix. 21; Vors. i 2 , p. 105. 26) says that Parmenides 'heard' 
Xenophanes but did not follow him. 

2 Diog. L. ix. 21 ( Vors. i 2 , p. 105. 29). * 

3 Proclus, In Parm. i, ad init. ( Vors. i 2 , p. 106. 30). 

4 Strabo, vi. 1. 1, p. 252 ( Vors. i 2 , p. 107. 39). 

5 Aristotle, Phys. iii. 6, 207 a 16 ; Fr. 8, line 42 {Vors. i 2 , p. 121. 3). 

6 Aristotle, De caelo iii. 1, 298 b 14; Aet. i. 24. 1 (D.G. p. 320; Vors. i 2 , 
p. no. 18). 

7 Fr. 8. 50-53 {Vors. i 2 , p. 121. 11-13). 

8 Tannery, op. cit., p. 123. 



64 PARMENIDES part ι 

this agrees with their notion that the universe breathes} a supposition 
which Tannery attributes to the Master himself because Xenophanes 
is said to have denied it. 2 Parmenides, on the other hand, denied 
the existence of the infinite void, and was therefore obliged to 
make his finite sphere motionless, and to hold that its apparent 
rotation is only an illusion. 3 

As in other respects the cosmology of Parmenides follows so 
closely that of the Pythagoreans, it is not surprising that certain 
astronomical innovations are alternatively attributed to Parmenides 
and to Pythagoras. Parmenides is said to have been the first to 
assert that the earth is spherical in shape and lies in the centre ; 4 
this statement has the great authority of Theophrastus in its favour ; 
there was, however, an alternative tradition stating that it was 
Pythagoras who first called the heaven κόσμος, and held the earth 
to be round (στρογγνλην). 5 As the idea that the earth is spherical 
was probably suggested by mathematical considerations, Pythagoras 
is the more likely to have conceived it, though Parmenides may 
have been the first to state it publicly (the Pythagorean secrecy, 
such as it was, seems to have applied only to their ritual, not to their 
mathematics or physics). Parmenides is associated with Democritus 
as having argued that the earth remains in the centre because, 
being equidistant from all points (on the sphere of the universe), 
it is in equilibrium, and there is no more reason why it should 
tend to move in one direction than in another. 6 Parmenides 
therefore here practically repeats the similar argument used by 
Anaximander (see above, p. 24), and we shall find that in other 
physical portions of his system he follows Anaximander and other 
Ionians pretty closely. 

1 Aristotle, Phys. iv. 6, 213 b 24. 

2 Tannery, op. cit., p. 121. Zeller (i 5 , p. 525), however, does not believe that 
the remark μη μίντοι άναπνάν, if Xenophanes really made it, is directed against 
the Pythagorean view. He points out, too, that the statement in Diog. L. ix. 19 
{Vors. i 2 , p. 34. 18), so far as these words ('but that it does not breathe') are 
concerned, may only represent an inference from the fact that Fr. 24 only 
mentions seeing, hearing, and thinking. This, however, assumes greater intelli- 
gence on the part of Diogenes than we are justified in attributing to him. 

3 Tannery, op. cit., p. 125. 

4 Diog. L. ix. 21 ( Vors. i 2 , p. 105. 32). 

5 Diog. L. viii. 48 ( Vors. i 2 , p. in. 38). 

6 Aet. iii. 15. 7 (D. G. p. 380 ; Vors. i 2 , p. m. 40) ; cf. Aristotle, De caelo, ii. 
13, 295 b 10, and the similar views in Plato, Phaedo 108 E-109 a. 



CH.ix PARMENIDES 65 

Secondly, Parmenides is said to have been the first to 'define 

the habitable regions of the earth under the two tropic zones ' ; Ύ 

on the other hand we are told that Pythagoras and his school 

declared that the sphere of the whole heaven was divided into five 

circles which they called ' zones \ 2 Hultsch 3 bids us reject the 

attribution to Pythagoras on the ground that these zones would 

only be possible on a system in which the axis of the universe 

about which it revolves passes through the centre of the earth ; 

the zones are therefore incompatible with the Pythagorean system, 

according to which the earth moves round the central fire. 

Hultsch admits, however, that this argument does not hold if the 

hypothesis of the central fire was not thought of by any one before 

Philolaus ; and there is no evidence that it was. As soon as 

Pythagoras had satisfied himself that the universe and the earth 

were concentric spheres, the centre of both being the centre of the 

earth, the definite portion of the heaven marked out by the extreme 

deviations of the sun in latitude (north and south) might easily 

present itself to him as a zone on the heavenly sphere. The Arctic 

Circle, already known in the sense of the circle including within 

it the stars which never set, would make another division, while 

a corresponding Antarctic Circle would naturally be postulated 

by one who had realized the existence of antipodes. 4 With the 

intervening two zones, five divisions of the heaven were ready to 

hand. It would next be seen that straight lines drawn from the 

centre of the earth to all points on all the dividing circles in the 

heaven would cut the surface of the earth in points lying on exactly 

corresponding circles, and the zone-theory would thus be transferred 

to the earth. 5 We are told, however, that Parmenides* division of 

the earth into zones was different from the division which would 

be arrived at in this way, in that he made his torrid zone about 

1 Aet. iii. 11. 4 (D. G. p. 377). 

2 Aet. ii. 12. 1 (D. G. p. 340). 

3 Hultsch, art. 'Astronomic' in Pauly-Wissowa's Real-Encyclopddie der 
classischen Altertwnswissenschaft, ii. 2, 1896, p. 1834. 

4 Alexander Polyhistor in Diog. L. viii. I. 26. 

5 Aet. iii. 13. I (D.G. p. 378), 'Pythagoras said that the earth was divided, 
correspondingly to the sphere of the universe, into five zones, the arctic, antarctic, 
summer and winter zones, and the equatorial zone; the middle of these defines 
the middle portion of the earth, and is for this reason called the torrid zone ; then 
comes the habitable zone which is temperate.' 

1410 F 



66 PARMENIDES parti 

twice as broad as the zone intercepted between the tropic circles, 
so that it spread over each of those circles into the temperate zones. 1 
This seems to be the first appearance of zones viewed from the 
standpoint of physical geography. 

Thirdly, Diogenes Laertius says, on the authority of Favorinus, 
that Parmenides is thought to have been the first to recognize that 
the Evening and the Morning Stars are one and the same, while 
others say that it was Pythagoras. 2 In this case, although 
Parmenides may have learnt the fact from the Pythagoreans, it 
is probable that Pythagoras did not know it as the result of 
observations of his own, but acquired the information from Egypt 
or Chaldaea along with other facts about the planets. 3 

On the purely physical side Parmenides in the main followed 
one or other of the Ionian philosophers. The earth, he said, was 
formed from a precipitate of condensed air. 4 He agreed with 
Heraclitus in regarding the stars as ' compressed ' fire (literally 
close-pressed packs of fire, πιλήματα πυρός). 5 

Parmenides* theory of ' wreaths' (στεφάναι) seems to be directly 
adapted from Anaximander's theory of hoops or wheels. Anaxi- 
mander had distinguished hoops belonging to the sun, the moon, 
and the stars respectively, which were probably concentric with 
the earth ; the hoops were of different sizes, the sun's being the 
largest, the moon's next, and those of the stars smaller still. These 
hoops were rings of compressed air filled with fire which burst out 
in flame at outlets, thereby producing what we see as the sun, 
moon, and stars. The corresponding views of Parmenides are not 
easy to understand ; I will therefore begin by attempting a transla- 
tion of the passage of Aetius in which they are set out. 6 

' There are certain wreaths twined round, one above the other 
[relatively to the earth as common centre] ; one sort is made of the 
rarefied (element), another of the condensed ; and between these 
are others consisting of light and darkness in combination. That 

1 Posidonius in Strabo, ii. 2. 2, p. 94. 

2 Diog. L. ix. 23 (Vors. i 2 , p. 106. 11). 

3 Tannery, op. cit., p. 229. 

4 Ae-yei θβ την γήν πυκνού καταρρνίντος aepos yeyovevai, Ps. Plllt. Stromat. 5 
{D. G. p. 581.4; Vors. i 2 , p. 109. 1). 

6 Aet. ii. 13. 8 (D. G. p. 342; Vors. i 2 , p. in. 25). Cf. Anaximander's 

πιλήματα άίρος. 

6 Aet. ii. 7. 1 (I). G. p. 335 ; Vors. i 2 , p. ill. 5-16). 



CH.ix PARMENIDES 67 

which encloses them all is solid like a wall, below which is a wreath 
of fire ; that which is in the very middle of all the wreaths is solid, 
about which (wept 0) [under which (νφ' ω, Diels)] again is a wreath 
of fire. And of the mixed wreaths the midmost is to all of them 
the beginning and cause of motion and becoming, 1 and this he calls 
the Deity which directs their course and holds sway (κληρονχον) 2 
[holds the keys (κλειδούχοι/, Fulleborn)], namely Justice and Necessity. 
Moreover, the air is thrown off the earth in the form of vapour 
owing to the violent pressure of its condensation ; the sun and the 
Milky Way are an exspiration 3 of the fire ; the moon is a mixture 
of both elements, air and fire. And, while the encircling aether 
is uppermost of all, below it is ranged that fiery (thing) which we 
call heaven, under which again are the regions round the earth.' 

But in addition we are told that 

' It is the mixture of the dense and the rarefied which produces 
the colour of the Milky Way.' 4 

1 The sun and the moon were separated off from the Milky Way, 
the sun arising from the more rarefied mixture which is hot, and 
the moon from the denser which is cold.' 5 

The fragments of Parmenides do not add much to this. The 
relevant lines are as follows : 

* The All is full of light and, at the same time, of invisible 
darkness, which balance each other ; for neither of them has any 
share in the other.' 6 

* Thou shalt learn the nature of the aether and all the signs in 
the aether, the scorching function of the pure clear sun, and whence 
they came ; thou shalt hear the wandering function and the nature 
of the round-eyed moon, and thou shalt learn of the surrounding 
heaven, whence it arose, and how Necessity, guiding it, compelled 
it to hold fast the bounds of the stars.' 7 

' (I will begin by telling) how the earth, the sun and the moon, 
the common aether, the milk of the heaven, furthest Olympus, and 
the hot force of the stars strove to come to birth.' 8 

1 I follow the reading adopted by Diels in the Vorsokratiker, άπάσαιε (άρχψ} 
re και (αΐτ'ιαν) κινήσεως καί yei/eaecu? νπάρχςιν. 

2 Burnet (Early Greek Philosophy, p. 219) observes that κλήρος in the Myth 
of Er suggests κληρουχον as the right reading. Fulleborn suggested κλτ)δοΰχον 
in view of the use of κλψδαε (keys) in Fr. 1. 14. 

3 The word αναπνοή is of course ambiguous ; I follow Diels' interpretation, 
' Ausdunstung ', ' evaporation ' or ' exhalation '. Diels (Parmenides Lehrgedichty 
1897, p. 105) compares άναπνοας Ισχον in the Timaeus 85 A. 

4 Aet. iii. 1. 4 (D. G. p. 365). 

5 Aet. ii. 20. 8 a (D. G. p. 349 ; Vors. i 2 , p. in. 35). 

6 Fr. 9 ( Vors. i 2 , p. 122. n-12). 

7 Fr. 10 (Vors. i 2 , pp. 122. 21-123. 2). 

8 Fr. n (Vors, i 2 , p. 123. 5-7). 

F 2 



68 PARMENIDES parti 

Of the wreaths he says that 

'The narrower (wreaths) were filled with unmixed 1 fire; those 
next in order to them (were filled) with night, and along with them 
the share of flame spreads itself. In the middle of these is the 
Deity which controls all/ 2 

It is not surprising that there have been a number of interpreta- 
tions of these passages taken in combination. 3 To begin with the 
outside, there is a doubt as to the relative positions of the ' heaven ' 
and the aether. According to Aetius 'the encircling aether is 
uppermost of all, and below it is ranged that fiery thing which 
we call heaven ', whereas the fragments suggest that the ' common 
aether' is within the 'encircling heaven* or 'furthest Olympus', 
which latter clearly seems to be the solid envelope compared to 
a wall. The fragments presumably better represent Parmenides' 
own statement, and possibly Aetius's version (which seems practi- 
cally to interchange the ' heaven ' and the * aether ') is due to some 
confusion. 

The next question is, what was the shape of the ' wreaths ' or 
bands? 4 Zeller, in view of the spherical form of the envelope, 
does not see how they can be anything but hollow globes. 5 But 
surely ' wreaths ' or ' garlands ', i. e. bands, would not in that case 
be a proper description. Tannery 6 takes them to be cylindrical 
bands fixed one inside the other, comparing with our passage the 
description in Plato's Myth of Er, 7 where ' the distaff of Necessity 
by means of which all the revolutions of the universe are kept up ' 
distinctly suggests that Plato had Parmenides' system in mind ; 
Plato there speaks of eight whorls (σφόνδυλοι), one inside the other, 
'like those boxes which fit into one another,' and of the lips of 

1 Reading άκρήτοιο. The reading άκρίτοιο (literally ' confused ' or ' undistin- 
guishable', that is to say, diluted fire) is impossible, because (i) it does not give 
the required sense, and (2) it offends against prosody, since t in άκριτος is short 
(Diels, Parmenides Lehrgedicht^ p. 104). 

2 Fr. 12 (Vors. i 2 , p. 123. 18-20). 

3 Zeller (i 5 , p. 573) gives references to the explanations suggested by Brandis, 
Karsten, and Krische. More recent views (those of Tannery, Diels, Berger, 
and Otto Gilbert) are referred to in the text above. 

4 στεφάνη is sometimes translated as 'crown'; but this rendering is open to 
the objection of suggesting a definite shape. Moreover, it is inapplicable to 
a series of wreaths or bands entwined the one within the other. 

5 Zeller, i 5 , p. 572. 6 Tannery, op. cit., p. 230. 
7 Plato, Republic x. 616 D. 



ch.ix PARMENIDES 69 

the whorls. In the Timaeus too there are no spheres, but bands 
or strips crossing one another at an angle. 1 We may perhaps take 
the bands to be, not cylinders, but zones of a sphere bisected by 
a great circle parallel to the bounding circles. Burnet 2 thinks that 
the solid circle which surrounds all the bands cannot be a sphere 
either, because in that case ' like a wall ' would be inappropriate. 
I do not, however, see any real difficulty in such a use of ' like 
a wall ', and certainly Parmenides' All was spherical. 3 

We now come to the main question of the nature of the bands, 
their arrangement relatively to one another, and the meaning to 
be attached to them severally. What we learn about them from 
Aetius and the fragments taken together amounts to this. First, 
the material of which they are composed is of two kinds ; one is 
alternatively described as the ' rarefied ' (άραιον), light, flame (φλόξ) 
or fire ; the other as the ' condensed ' (πνκνόν), darkness, or night. 
The bands are of three kinds, the first composed entirely of the 
1 rarefied ' element or fire, the second of the ' condensed ' or darkness, 
and the third of a mixture of the two. Secondly, as regards their 
arrangement, we are told that there is a solid envelope, a spherical 
shell, enclosing them all ; two bands of unmixed fire are mentioned, 
of which one is immediately under the envelope, the other is about 
(reading nepi with the MSS.) or under (reading υπό with Diels) 
* that which is in the very midst of all the bands ' and which is 
1 solid ' ; these two bands are also ' narrower ' (than something), 
where ' narrower ' means that their radii are smaller, that is to say, 
their inner surfaces are nearer (than something) to the centre of 
the earth, which is the common centre of all the bands. The mixed 
bands, according to Aetius, are ' between ' the bands of fire and the 
bands of darkness ; the fragment ( 1 2) makes them come next to 
both the ' narrower ' bands, the bands of fire. 

There seems to be general agreement that the ' mixed ' bands 
include the sun, the moon, and the planets ; it is with regard to 
the meaning and position of the bands of fire, and to the place 
occupied by the Deity called by the names of Justice and Necessity, 
that there has been the greatest difference of opinion. Tannery's 

1 Plato, Timaeus 36 Β. 2 Burnet, Early Greek Philosophy, p. 216. 

3 ' It is complete on every side, like the mass of a well-rounded sphere poised 
from the centre in every direction' (Fr. 8. 42-4; Vors. i 2 , p. 121. 3-5). 



ηο PARMENIDES parti 

view is that the outermost band of fire under the solid envelope (which 
envelope may be regarded as one of the bands made of the ' condensed' 
element) is the Milky Way. In that case, however, the fire is not 
pure ; for ' it is the mixture of the dense and the rarefied which 
produces the colour of the Milky Way '. Tannery would get over 
this difficulty by supposing the band to be only full of fire, like 
the hoops of Anaximander, the almost continuous brightness being 
due to exspiration through the covering. But Aetius says that both 
the Milky Way and the sun are an exspiration of fire, and the sun 
is certainly represented by one of the mixed bands, so that the 
Milky Way should also be one of the mixed bands. The band 
of fire which (with the reading πβρΓ) is about the solid in the very 
centre of all the bands (i. e. the earth) Tannery takes to be our 
atmosphere. This seems possible, for Parmenides may have re- 
garded air lit up as being fire. In Diels' interpretation a similar 
view seems to be taken of the outermost band of fire which he 
calls ' aether-fire ' ; and the assumption that the aether is fire is 
perhaps justified by the fact, if true, that Parmenides declared the 
heaven to be of fire. 1 The intermediate bands consisting of the 
two elements, light and dark, in combination correspond in Tannery's 
view to the orbits of the moon, the sun, and the planets respectively, 
which (starting from the earth) come in that order ; possibly among 
these mixed bands there may be bands entirely dark as well (cf. 
Fr. 12). 

Diels 2 takes the bands which consist exclusively of the 'condensed' 
element to be made of earth simply. There are two of these ; 
one is the solid envelope, the solid firmament, ' Outer Olympus ' ; 
the other is the crust of the earth. Just beneath the solid envelope 
comes the outer band of fire, which is the aether-fire. Next within 
this come the mixed class of bands which are the bands of stars 
containing both elements, earth and fire, not separate from one 
another but mixed together. Such dark rings, out of which the 
fire flashes out here and there, are the Milky Way, the sun, the 
moon, and the planets. After the mixed bands comes the solid 
earth-crust, below which again (reading νφ' ω, which Diels substitutes 

1 Aet. ii. II. 4 (D. G. p. 340 ; Vors. i 2 , p. ill. 23). 

2 Diels, Vors. ii 2 , i, p. 675 ; cf. Parmenides Lehrgedicht, 1897, pp. 104 sqq. 



ch.ix PARMENIDES 71 

for πβρι ο) comes the inner band of fire, which therefore is inside 
the earth and forms a kernel of fire. 

It will be seen that the idea of Anaximander that stars are dark 
rings with fire shining out at certain points is supposed, both by 
Tannery and Diels, to be more or less present in Parmenides' con- 
ception, though Tannery only assumes it as applying to the Milky 
Way, which he wrongly identifies with the outer band of undiluted 
fire. Diels, more correctly, implies that it is the mixed rings made 
up of light and darkness in combination which exhibit the pheno- 
menon of ' fire shining out here and there ', these mixed rings 
including the Milky Way as well as the sun, moon, and planets. 
It is possible that Aetius's ' mixed rings ' may be no more than 
his interpretation of the line in Fr. 12 which says that after the 
'narrower' bands 'filled with unmixed fire' there come 'bands 
filled with night and with them (μβτά, which Diels translates by 
' between ') is spread (or is set in motion, i'trcci) a share of fire '. 
And this line itself may mean either that the bands of night have 
a portion of fire mixed in them, or that each of the bands of night 
has a stream of fire (its ' share of fire ') coursing through it. If the 
fire were enclosed in the darkness as under the second alternative, 
we should have a fairly exact reproduction of Anaximander's tubes 
containing fire ; but there is nothing in the fragment to suggest 
that fire shines out of vents in the dark covering ; hence the mixture 
of light and dark, with light shining out at certain points (without 
enclosure in tubes), as assumed by Diels, seems to be the safer 
interpretation. 

Tannery and Diels differ fundamentally about the inner band 
of fire. According to the former, it is the atmosphere round the 
earth, and, if the ' atmosphere ' be taken to include the empty space 
outside the actual atmosphere as far as the nearest of the mixed 
bands, this seems quite possible. Diels, however, (reading ύψ ω, 
' under which ', instead of nepl 0, ' round which '), makes it a kernel 
of fire inside the earth and concludes that ' Parmenides is for us the 
first who stated the truth not only as regards the form of the earth 
but also as regards its constitution, whether he guessed the latter 
or inferred it correctly from indications such as volcanoes and hot 
springs \ l But it seems to me that there are great difficulties in 
1 Diels, Parmenides Lehrgedicht, pp. 105, 106. 



7 2 PARMENIDES parti 

the way of Diels' interpretation. First, it is difficult to regard 
a kernel of fire, which would presumably be a solid mass of fire, 
spherical in shape, as satisfying the description of a wreath or band. 
Secondly, whereas Fr. 12 speaks of the narrower bands as filled 
with unmixed fire and then of the mixed bands being ' next to 
these ' (ai δ' km ταΐς νυκτός . . .), the mixed bands would, on Diels' 
interpretation, be next to only one of the bands of fire (the outer 
one) and would not be next to the inner one but would be separated 
from it by the earth's crust. Diels seems to have anticipated this 
objection, for he explains that it is both the unmixed kinds of bands 
(i.e. those made of unmixed fire and those of unmixed earth, and 
not only the former, the 'narrower' bands) on which the mixed 
bands follow, in the inward direction starting from the outside 
envelope and in the outward direction starting from the centre ; * 
but ταΐς would much more naturally mean the narrower bands 
only. Thirdly, it seems to me to be difficult to assume that there 
is no band intervening between the surface of the earth and the 
nearest of the mixed bands ; if there were no intervening band, 
the nearest mixed band, say that of the moon, would have to be 
in contact with the earth, and therefore the moon also, shining out 
of it, must practically touch the earth. Therefore there must be 
some intervening band. But, if there is an intervening band, it 
must be one of three kinds, dense, mixed, or fiery. It cannot 
be a dense band, for, if it were, the sun, moon and stars would never 
be visible ; if it were a mixed band, there would again be some 
heavenly body or bodies in the same position of virtual contact 
with the earth ; therefore the intervening band can only be a band 
of fire. I am disposed, therefore, to accept Tannery's view that the 
inner band of fire is our atmosphere with the empty space beyond 
it reaching to the mixed bands. 

If the above arguments are right, the order would be, starting 
from the outside: (1) the solid envelope like a wall; (2) a band 
of fire = the aether-fire ; (3) mixed bands, in which are included 
the Milky Way, the planets, the sun, and the moon ; (4) a band 
of fire, the inner side of which is our atmosphere, touching the 
earth; (5) the earth itself; which is Diels' solution except as 
regards (4). 

1 Parmenides Lehrgedicht, p. 106. 



cii.ix PARMENIDES 73 

Berger 1 has an ingenious theory as regards the inner band of 
fire round the earth. If I understand him rightly, he argues that 
the bands in the heaven containing the stars were described in one 
part of Parmenides' poem, and the zones of the earth in another, 
and that Fr. 12 refers to the zones; that the two descriptions then 
got confused in the Doxographi, and that the inner band of fire 
is really nothing but the torrid zone, which has no business in the 
description at all. Diels has shown that this cannot be correct. 2 
Gilbert 3 disagrees with Diels' view of the inner band of fire as 
a kernel of fire inside the earth ; he himself thinks that there was 
not a band of fire about the earth, but that πνρώδης (with στεφάνη 
understood), ' a band of fire ', is a mistake for πυρ, ' fire ', or πνρωδς? 
in the neuter, and that the meaning is a fire or a fiery space 
connected with the earth (nepi in that sense being possible) ' down- 
wards ', which fire or fiery space he says we must suppose to 
embrace the under surface of the earth's sphere. 

Lastly, there is a difficulty as to the position occupied by the 
1 goddess who steers all things ', Justice or Necessity. This mytho- 
logical personification of Necessity and Justice is, of course, after the 
Pythagorean manner, 4 and reminds us of the similar introduction 
of Necessity in Plato's Myth of Er, which has so many other points 
of resemblance to Parmenides' theory. Fragment 12 says that this 
Deity is 'in the middle of these', i.e. presumably 'these bands', and 
Aetius, that is to say Theophrastus, took this to mean in the midst 
of the ' bands filled with night but with a share of fire in them '. 
Simplicius, on the other hand, takes it to mean ' in the middle of the 
whole system (kv μέσω πάντων) ', 5 i. e. in the middle of the whole world, 
clearly identifying the goddess with the central fire or hearth of the 
Pythagoreans. Diels seems to favour Simplicius's view, taking the 
centre of the universe to be the centre of the earth, 6 without, how- 

1 Berger, Geschichte der wissenschaftlichen Erdkunde der Griechen, p. 204 sq. 

2 Parmenides Lehrgedicht, p. 104. Since the torrid zone, as viewed by Par- 
menides, is twice the size of the zone between the tropics, the ' narrower ' zones 
must be the temperate zones, which requires the impossible reading άκρίτοιο ; 
with the true reading άκρήτοιο, the torrid zone would be ' broader ', not 
' narrower '. Besides, Aetius's paraphrase agrees so closely with the fragment, 
especially in the striking introduction of the Deity, that it cannot be regarded 
as being anything else than Theophrastus's paraphrase of the verses. 

3 Gilbert, 'Die δαίμων des Parmenides', in Archiv fur Gesch. der Philosophie, 
xx, 1906, pp. 25-45. 4 Tannery, loc. cit. 

5 Simpl. in Phys. p. 34. 15 ( Vors. i 2 , p. 123. 16). 

6 Diels, Parmenides Lehrgedicht, pp. 107-8. 



74 PARMENIDES part ι 

ever, attempting to reconcile this with Aetius's statement that she 
is placed in the middle of the mixed bands. It is in any case 
difficult to suppose that Parmenides treated his goddess who 
* guides the encircling heaven and compels it to hold fast the 
bounds of the stars ' as shut up within a solid spherical earth with 
no outlet ; the difficulty is even greater than in the Myth of Er, 
where at all events there is ' a straight light like a pillar which 
extends from above through all the heaven and earth ', and which 
accordingly passes through the place where Necessity is assumed 
to be seated. The statement of Aetius that she is placed in the 
middle of the mixed bands suggested to Berger 1 the possibility 
that her place was in the sun, in view of the pre-eminent position 
commonly assigned to the sun in the celestial system. 2 Gilbert 
holds that the goddess had her abode in the fiery space under the 
earth above mentioned ; he quotes from other poets, Hesiod, 
Heraclitus, Aeschylus and Sophocles, references to Dike as con- 
nected with the gods of the lower world, his object being to show 
that, in connecting Justice or Necessity with the earth, night, 
and the under-world, Parmenides was only adopting notions 
generally current. 3 Gilbert (like Diels) is confronted with the 
difficulty of Aetius's location of the goddess ■ in the middle of 
the mixed bands! and he disposes of this objection by assuming that 
the words were interpolated by some one who wished to find her in 
the sun. 4 This, however, seems too violent. 

Both Tannery and Diels specially mention the planets, and Tannery 
makes Parmenides arrange the heavenly bodies in the following order, 
starting from the earth : moon, sun, planets, fixed stars. There 
is, however, nothing in the texts about the bands which distinguishes 
the planets from the fixed stars or indicates their relative distances. 

1 Berger, op. cit, pp. 204, 205. 

2 e. g. Cleanthes (Aet. ii. 4. 16) saw in the sun the seat of authority in the 
universe (το ηγεμονικον τον κόσμου) : cf. also such passages as Theon of Smyrna, 
pp. 138. 16, 140. 7, 187. 16; Plut. Defac. in orbe lunae 30, 945 c; Proclus, in 
Timaeum 258 A, 'The sun, where the justice ordering the world is placed.' 

3 Gilbert, loc. cit., p. 36. 

4 The text in Diels' Doxographi (p. 335. 10 sq.) being και το μεσαίτατον 
πασών περί ο πάλιν πνρωδης' των δε συμμιγων την μεσαιτάτην άπάσαις τοκεα πάσης 
κινήσεως και γενέσεως υπάρχειν, ηντινα κα\ δαίμονα κ.τ.έ., Gilbert would reject των dt 
σνμμιγών τήν μεσαιτάτην as an interpolation, leaving κα\ το μεσαίτατον πασών, 
περί ο πάλιν πνρωδης (? στεφάνη), άπάσαις τοκεα πάσης κινήσεως και γενέσεως 
υπάρχειν κ.τ.ε. 



ch.ix PARMENIDES y$ 

The only passage in the Doxographi throwing light on the matter 
is a statement that 

1 Parmenides places the Morning Star, which he thinks the same 
as the Evening Star, first in the aether ; then, after it, the sun, and 
under it again the stars in the fiery (thing) which he calls heaven.' τ 

Tannery thinks that, if Parmenides distinguished Venus, and 
if it was from the first Pythagoreans that he learnt to do so, the 
other planets must equally have been known to the Pythagoreans 
and therefore to Parmenides. Tannery's view, however, of 
Parmenides' arrangement of the stars can hardly be reconciled 
with the distinct statement of Aetius that, while Venus is outside 
the sun, the other stars are below it ; this, except as regards Venus, 
agrees with Anaximander's order, according to which both the 
planets and the other stars are all placed below the sun and moon. 
Tannery is therefore obliged to assume that Aetius's remark is an 
error based on a too rigorous interpretation of the terms aether 
and heaven ; this, however, seems somewhat arbitrary. 

It remains to deal with the statement of the Doxographi that 
Parmenides held the moon to be illuminated by the sun : 

* The moon Parmenides declared to be equal to the sun ; for 
indeed it is illuminated by it.' 2 

This is the more suspicious because in another place Aetius 
attributes the first discovery of this fact to Thales, and adds that 
Pythagoras, Parmenides, and Empedocles, as well as Anaxagoras 
and Metrodorus, held the same view. 3 Parmenides was doubtless 
credited with the discovery on the ground of two lines from his 
poem. 4 The first is quoted by Plutarch : 5 

' For even if a man says that red-hot iron is not fire, or that the 
moon is not a sun because, as Parmenides has it, the moon is 

11 a night-shining foreign light wandering round the earth ", 

he does not get rid of the use of iron or of the existence of the 



moon.' 



1 Aet. ii. 15. 7 (D. G. p. 345). 

2 Aet. ii. 26. 2 (D. G. p. 357 ; Vors. i 2 , p. ill. 32). 

3 Aet. ii. 28. 5 (D. G. p. 358 ; Vors. i 2 , p. ill. 33). 

4 Fr. 14 and 15 (Vors. i 2 , p. 124. 6, 10). 

5 Plutarch, Adv. Colot. 15, p. 11 16 A (Vors. i 2 , p. 124, 4-7). 



ηβ PARMENIDES parti 

But, even if the verse is genuine, ' foreign ' (άλλοτριον) need not 
have meant ' borrowed ' ; the expression άλλότρων φώ? is, as Diels 
says, 1 a witty adaptation of Homer's αλλότριος φώ? used of persons, 
' a stranger '. 2 Tannery thinks that the line is adapted from one 
of Empedocles', and was probably interpolated in Parmenides' 
poem by some Neo-Pythagorean who was anxious to refer back 
to the Master the discovery which gives Anaxagoras his greatest 
title to fame. 3 

Boll, 4 on the other hand, considers it absolutely certain that 
Parmenides knew of the illumination of the moon by the sun. 
He admits, however, that we cannot suppose Parmenides to 
have discovered the fact for himself, and that we cannot be 
certain whether he got it from Anaximenes or the Pythagoreans. 
We have seen (p. 19) good reason for thinking that it was not 
Anaximenes who made the discovery; and the only support that 
Boll can find for the alternative hypothesis is the statement of 
Aetius that Pythagoras considered the moon to be a ' mirror-like 
body ' (κατοπτροςίδβς σώμα). 5 But this is an uncertain phrase to 
build upon, especially when account is taken of the tendency to 
attribute to Pythagoras himself the views of later Pythagoreans ; 
and indeed the evidence attributing the discovery to Anaxagoras 
is so strong that it really excludes all other hypotheses. 

The other line speaks of the moon as ' always fixing its gaze 
on the beams of the sun \ This remark is certainly important, 
but is far from explaining the cause of the observed fact. But 
we have positive evidence against the attiibution of the discovery 
of the opacity of the moon to Parmenides or even to Pythagoras. 
It is part of the connected prose description of Parmenides' 
system 6 that the moon is a mixture of air and fire ; 7 in other 
passages we are told that Parmenides held the moon to be of fire 8 

1 Diels, Vors. ii 2 . 1, p. 675 ; Parmenides Lehrgedicht, p. no. 

2 Homer, Iliad v. 214 ; Od. xviii. 219, &c. 

3 Tannery, op. cit., p. 210. The lines are respectively — 

Νυκτιφα€5 π*ρ\ γαϊαν άλώμςνον αλλότριοι' φως (Parm.). 
Κυκλοτ€ρ€ς π(ρ\ yaiav iXiaaerai άλλότρων φως (Emped.). 

4 Boll, art. ' Finsternisse ' in Pauly-Wissowa's Real-Encydopddie der classischen 
Altertumswissenschaft, vi. 2, 1909, p. 2342. 

5 Aet. ii. 25. 14 (D. G. p. 357). 

6 Aet. ii. 7. 1 (D. G. p. 335 ; Vors. i 2 , p. III. 5 sqq.). 

7 Ibid. {D. G. p. 335 ; Vors. i 2 , p. in. 13). 

8 Aet. ii. 25. 3 (D. G. p. 356 ; Vors. i 2 , p. in. 31). 



ch.ix PARMENIDES 77 

and to be an excretion from the denser part of the mixture in the 
Milky Way, 1 which itself (like the sun) is an exspiration of fire. 2 
More important still is the evidence of Plato, who speaks of ' the 
fact which Anaxagoras lately asserted, that the moon has its light 
from the sun \ 3 It seems impossible that Plato should have spoken 
in such terms if the fact had been stated for the first time by 
Parmenides or the Pythagoreans. 

1 Aet. ii. 20. 8 a (D. G. p. 349 ; Vors. i 2 , p. 1 1 1. 35). 

2 Aet. ii. 7. I (D. G. p. 335 ; Vors. i 2 , p. ill. 13). 

3 Plato, Cratylus 409 A. 



χ 

ANAXAGORAS 

Anaxagoras was born at Clazomenae in the neighbourhood of 
Smyrna about 500 B.C. He neglected his possessions, which were 
considerable, in order to devote himself to science. 1 Some one once 
asked him what was the object of being born, to which he replied, 
' The investigation of sun, moon, and heaven.' 2 He seems to have 
been the first philosopher to take up his abode at Athens, where he 
enjoyed the friendship of Pericles, who had probably induced him to 
come thither. When Pericles became unpopular shortly before the 
outbreak of the Peloponnesian war, he was attacked through his 
friends, and Anaxagoras was accused of impiety for holding that 
the sun was a red-hot stone and the moon earth. 3 According to 
one account he was fined five talents and banished; 4 another 
account says that he was put in prison and it was intended to put 
him to death, but Pericles got him set at liberty ; 5 there are other 
variations of the story. He went and lived at Lampsacus, where he 
died at the age of 72. 

A great man of science, Anaxagoras enriched astronomy by one 
epoch-making discovery. This was nothing less than the discovery 
of the fact that the moon does not shine by its own light but 
receives its light from the sun. As a result, he was able to give 
(though not without an admixture of error) the true explanation of 
eclipses. I quote the evidence, which is quite conclusive : 

' . . . the fact which he (Anaxagoras) recently asserted, namely 
that the moon has its light from the sun.' 6 

1 Now when our comrade, in his discourse, had expounded that 
proposition of Anaxagoras, that " the sun places the brightness in 
the moon ", he was greatly applauded.' 7 

1 Plato, Hippias Major 283 A. 2 Diog. L. ii. 10 {Vors. i 2 , p. 294. 17). 

3 Plato, Apology 26 D. 4 Diog. L. ii. 12 ( Vors. i 2 , p. 294. 32). 

6 Ibid. ii. 13 {Vors. i 2 , p. 294. 42). 6 Plato, Cratylus, p. 409 A. 

7 Plutarch, De facie i?i orbe lunae 16, p. 929 Β (Vors. i 2 , p. 321. 5-7). 



ANAXAGORAS 79 

i The moon has a light which is not its own, but comes from the 
sun. x 

' The moon is eclipsed through the interposition of the earth, 
sometimes also of the bodies below the moon' 2 [i.e. the 'bodies 
below the stars which are carried round along with the sun and the 
moon but are invisible to us '. 3 ] 

' The sun is eclipsed at the new moon through the interposition 
of the moon.' 4 ' He was the first to set out distinctly the facts 
about eclipses and illuminations.' 5 

' For Anaxagoras, who was the first to put in writing, most 
clearly and most courageously of all men, the explanation of the 
moon's illumination and darkness, did not belong to ancient times, 
and even his account was not common property but was still a 
secret, current only among a few and received by them with caution 
or simply on trust. For in those days they refused to tolerate the 
physicists and star-gazers as they were called, who presumed to 
fritter away the deity into unreasoning causes, blind forces, and 
necessary properties. Thus Protagoras was exiled, and Anaxa- 
goras was imprisoned and with difficulty saved by Pericles.' 6 

'Anaxagoras, in agreement with the mathematicians, held that 
the moon's obscurations month by month were due to its following 
the course of the sun by which it is illuminated, and that the 
eclipses of the moon were caused by its falling within the shadow 
of the earth, which then comes between the sun and the moon, 
while the eclipses of the sun were due to the interposition of the 
moon.' 7 

'Anaxagoras, as Theophrastus says, held that the moon was 
also sometimes eclipsed by the interposition of the (other) bodies 
below the moon.' 8 

Here,' then, we have the true explanation of lunar and other 
eclipses, though with the unnecessary addition that, besides the 
earth, there are other dark bodies invisible to us which sometimes 

1 Hippolytus, Refut. i. 8. 8 (from Theophrastus : see D. G. p. 562 ; Vors. i 2 , 
p. 301. 46). 

2 Ibid. i. 8. 9. {D. G. p. 562 ; Vors. i 2 , p. 301. 47). 

3 Ibid. i. 8. 6 (D. G. p. 562 ; Vors. i 2 , p. 301. 41). 

4 Ibid. i. 8. 9 {D. G. p. 562 ; Vors. i 2 , p. 301. 48). 

5 Ibid. i. 8. 10 {D. G. p. 562; Vors. i 2 , p. 302. 3). 

6 Plutarch, Nic. 23 {Vors. i 2 , p. 297. 40-6). 

7 Aet. ii. 29. 6 {D. G. p. 360; Vors. i 2 , p. 308. 17). I have in the last phrase 
translated Diels' conjecturally emended reading ήλιον de τής σελήνης instead of 
μάλλον 8e της σελήνης άντιφραττομένης {D. G. pp. 53-4). The difficulty, however, 
is that, according to the heading, the passage deals with the eclipses of the 
moon only. 

8 Aet. ii. 29. 7 (D. G. p. 360 ; Vors. i 2 , p. 308. 20). 



8o ANAXAGORAS part ι 

obscure the moon and cause eclipses. In this latter hypothesis, as 
in much else, Anaxagoras followed Anaximenes. 1 

Whether Anaxagoras reached the true explanation of the phases 
of the moon is much more doubtful. It is true that Parmenides 
had observed that the moon has its bright portion always turned in 
the direction of the sun ; when to this was added Anaxagoras's 
discovery that the moon derived its light from the sun, the explana- 
tion of the phases was ready to hand. But it required that the 
moon should be spherical in shape ; Anaxagoras, however, held 
that the earth, and doubtless the other heavenly bodies also, were 

1 The same idea is attributed by Aristotle {De caelo ii. 13, 293 b 21-25) to 
certain persons whom he does not name : ' Some think it is possible that more 
bodies of the kind [i.e. such as the Pythagorean counter-earth] may move about 
the centre but may be invisible to us owing to the interposition of the earth. 
This, they say, is the reason why more eclipses of the moon occur than of the 
sun, for each of the bodies in question obscures the moon, and it is not only the 
earth which does so.' An interesting suggestion has been made (by Boll in art. 
'Finsternisse' in Pauly-Wissowa's Real-Encyclopddie d. class. Alter tumsw. vi. 2, 
p. 2351), which furnishes a conceivable explanation of the persistence of the 
idea that lunar eclipses are sometimes caused by the interposition of dark bodies 
other than the earth. Cleomedes {De motu circulari ii. 6, p. 218. 8. sqq.) 
mentions that there were stories of extraordinary eclipses which 'the more 
ancient of the mathematicians ' had vainly tried to explain ; the supposed 
' paradoxical ' case was that in which, while the sun seems to be still above the 
horizon, the eclipsed moon rises in the east. The phenomenon appeared to be 
inconsistent with the explanation of lunar eclipses by the entrance of the moon 
into the earth's shadow ; how could this be if both bodies were above the 
horizon at the same time ? The ' more ancient ' mathematicians tried to argue 
that it was possible that a spectator standing on an emi?ience of the spherical 
earth might see along the generators of a cone, i.e. a little downwards on all 
sides, instead of merely in the plane of the horizon, and so might see both the 
sun and the moon even when the latter was in the earth's shadow. Cleomedes 
denies this and prefers to regard the whole story of such cases as a fiction 
designed merely for the purpose of plaguing astronomers and philosophers ; no 
Chaldean, he says, no Egyptian, and no mathematician or philosopher has 
recorded such a case. But we do not need the evidence of Pliny (N.H. ii, c. 57, 
§ 148) to show that the phenomenon is possible ; and Cleomedes himself really 
gives the explanation (pp. 222. 28-226. 3), namely, that it is due to atmospheric 
refraction. Observing that such cases of atmospheric refraction were especially 
noticeable in the neighbourhood of the Black Sea, he goes on to say that it is 
possible that the visual rays going out from our eyes are refracted through falling 
on wet and damp air, and so reach the sun though it is already below the 
horizon ; and he compares the well-known experiment of the ring at the bottom 
of a jug, where the ring, just out of sight when the jug is empty, is brought into 
view when water is poured in. Unfortunately there is nothing to indicate the 
date of the ' more ancient mathematicians ' who gave the somewhat primitive 
explanation which Cleomedes refutes ; but was it the observation of the phe- 
nomenon, and their inability to explain it otherwise, which made Anaxagoras 
and others adhere to the theory that there are other bodies besides the earth 
which sometimes, by their interposition, cause lunar eclipses ? 



ch. x ANAXAGORAS 81 

flat, and accordingly his explanation of the phases could hardly 
have been correct. 1 

Anaxagoras's cosmology contained other fruitful ideas. Accord- 
ing to him the formation of the world began with a vortex set up, 
in a portion of the mixed mass in which 'all things were together', 
by his dens ex machina, Nous. 2 This rotatory movement began at 
one point and then gradually spread, taking in wider and wider 
circles. The first effect was to separate two great masses, one 
consisting of the rare, hot, light, dry, called the ' aether ', and the 
other of the opposite categories and called 'air'. The aether or 
fire took the outer position, the air the inner. 3 The next step is the 
successive separation, out of the air, of clouds, water, earth, and 
stones. 4 The dense, the moist, the dark and cold, and all the 
heaviest things collect in the centre as the result of the circular 
motion ; and it is from these elements when consolidated that the 
earth is formed. 5 But, after this, ' in consequence of the violence of 
the whirling motion, the surrounding fiery aether tore stones away 
from the earth and kindled them into stars.' 6 Reading this with 
the remark that stones ' rush outwards more than water ', 7 we see 
that Anaxagoras conceived the idea of a centrifugal force as dis- 
tinct from that of concentration brought about by the motion of 
the vortex, and further that he assumed a series of projections or 
1 hurlings-ofT' of precisely the same kind as the theory of Kant and 
Laplace assumes for the formation of the solar system. 8 

Apart from the above remarkable innovations, Anaxagoras did 
not make much advance upon the crude Ionian theories ; indeed he 
showed himself in the main a follower of Anaximenes. 

According to Anaxagoras 

' The earth is fiat in form and remains suspended because of its 
size, because there is no void, and because the air is very strong and 
supports the earth which rides upon it.' 9 

' The sun, the moon, and all the stars are stones on fire, which 
are carried round by the revolution of the aether.' 10 

1 Tannery, op. cit., p. 278. s Fragment 13 (Vors. i 2 , p. 319. 20). 

8 Fr. 15 {Vors. i 2 , p. 320. 11). 4 Fr. 16 (Vors. i 2 , p. 320. 20). 

5 Hippol. Refut. i. 8. 2 (from Theophrastus) ; D. G. p. 562 ; Vors. 1 2 , p. 301. 
28-30. 6 Aet. ii. 13. 3 (D. G. p. 341 ; Vors. i 2 , p. 307. 16). 

7 Fr. 16 ( Vors. i 2 , p. 320. 22-3). 

8 Gomperz, Griechische Denker, i 3 , p. 176. 

9 Hippol. Refut. i. 8. 3 (D. G. p. 562 ; Vors. i 2 , p. 301. 31). 

10 Ibid. i. 8. 6 ( Vors. i 2 , p. 301. 39). 

1410 G 



8<z ANAXAGORAS part]i 

4 The sun is a red-hot mass or a stone on fire.' l 
* It is larger* (or ' many times larger' 2 ) than the Peloponnese. 3 
' The moon is of earthy nature and has in it plains and ravines.' 4 
1 The moon is an incandescent solid, having in it plains, moun- 
tains, and ravines.' 5 

' It is an irregular compound because it has an admixture of cold 
and of earth. It has a surface in some places lofty, in others low, 
in others hollow. And the dark is mixed along with the fiery, the 
joint effect being an impression of the shadowy ; hence it is that 
the moon is said to shine with a false light.' β 

Anaxagoras explained the 'turning' of the sun at the solstice 
thus: 

' The turning is caused by the resistance of the air in the north 
which the sun itself compresses and renders strong through its 
condensation.' 7 

1 The turnings both of the sun and of the moon are due to their 
being thrust back by the air. The moon's turnings are frequent 
because it cannot get the better of the cold.' 8 

Again : 

f We do not feel the warmth of the stars because they are at 
a great distance from the earth ; besides which they are not as hot 
as the sun because they occupy a colder region. The moon is 
below the sun and nearer to us.' 9 

' The stars were originally carried round (laterally) like a dome, 
the pole which is always visible being vertically above the earth, 
and it was only afterwards that their course became inclined.' 10 

' After the world was formed and the animals were produced 
from the earth, the world received as it were an automatic tilt 
towards its southern part, perha ps b y^clesign, in order that some 

1 Aet. ii. 20. 6 (D. G. p. 349; Vors. i 2 , p. 307. 19). 

2 Aet. ii. 21. 3 (D. G. p. 351 ; Vors. i 2 , p. 307. 20J. 

3 Diog. L. ii. 8 ( Vors. i 2 , p. 293. 38). 

4 Hippol. Refut. i. 8. 10 (D. G. p. 562 ; Vors. i 2 , p. 302. 4). 

5 Aet. ii. 25. 9 (D. G. p. 356 ; Vors. i 2 , p. 308. 10). 

6 Aet. ii. 30. 2 (D. G. p. 361 ; Vors. i 2 , p. 308. 12). As Dreyer observes 
{Planetary Systems, p. 32, note), the moon has some light of its own which we 
see during lunar eclipses ; cf. Olympiodorus on Arist. Meteor, (p. 67. 36, ed. 
Stiive; Meteor., ed. Ideler, vol. i, p. 200), 'The moon's own light is of one kind, 
the sun's of another ; for the moon's own light is like charcoal (άνθρακώδα), as 
we can plainly see during an eclipse.' 

7 Aet. ii. 23. 2 (D. G. p. 352 ; Vors. i 2 , p. 307. 20). 

8 Hippol. Refut. i. 8. 9 (D. G. p. 562 ; Vors. i 2 , p. 302. 1). 

9 Ibid. i. 8. 7 (D. G. p. 562 ; Vors. i 2 , p. 301. 42). 
10 Diog. L. ii. 9 (Vors. i 2 , p. 294. 3). 



ch. x ANAXAGORAS 83 

parts of the world might become uninhabitable and others inhabit- 
able, according as they are subject to extreme cold, torrid heat, 
or moderate temperature.' * 

1 The revolution of the stars takes them round under the earth.' 2 

Gomperz 3 finds a difficulty in reconciling the last of these 
passages with the other statement that the earth is flat and rests on 
air, in which Anaxagoras had followed Anaximenes. Anaximenes 
seems to have regarded the basis of air on which the flat earth 
rested in the same way as Thales the water on which his earth 
floated ; and Anaximenes said that the stars did not pass under the 
earth but laterally round it. I do not, however, feel sure that 
Anaxagoras could not have supposed the stars to pass in their 
revolution through the basis of air under the earth, although no 
doubt Thales was almost precluded from supposing them to pass 
through his basis of water. If, as Gomperz says, Simplicius 4 is alone 
in attributing to Anaxagoras's earth the shape of a drum or cylinder, 
Aristotle as well as Simplicius seems to imply that at all events 
the earth occupied the centre of the universe. 5 

Anaxagoras put forward a remarkable and original hypothesis to 
explain the Milky Way. As we have seen, he thought the sun to be 
smaller than the earth. Consequently, when the sun in its revolu- 
tion passes below the earth, the shadow cast by the earth extends 
without limit. The trace of this shadow on the heavens is the 
Milky Way. The stars within this shadow are not interfered with 
by the light of the sun, and we therefore see them shining ; those 
stars, on the other hand, which are outside the shadow are over- 
powered by the light of the sun, which shines on them even during 
the night, so that we cannot see them. Such appears to be the 
meaning of the passages in which Anaxagoras's hypothesis is 
explained. According to Aristotle, Anaxagoras and Democritus 
both held that 

1 The Milky Way is the light of certain stars. For when the sun 
is passing below the earth some of the stars are not within its 
vision. Such stars then as are embraced in its view are not seen to 

1 Aet. ii. 8. 1 {D. G. p. 337 ; Vors. i 2 , p. 306. 12). 

2 Hippol. Refut. i. 8. 8 (D. G. p. 562 ; Vors. i 2 , p. 301. 47). 

3 Gomperz, Griechische Denker, i 3 , pp. 1 yS t 442. 

4 Simplicius on De caelo, p. 520. 30. 
6 Arist. De caelo ii. 13, 295 a 13. 

G 2 



84 ANAXAGORAS parti 

give light, for they are overpowered by the rays of the sun ; such of 
the stars, however, as are hidden by the earth, so that they are not 
seen by the sun, form by their own proper light the Milky Way.' x 

'Anaxagoras held that the shadow of the earth falls in this part 
of the heaven (the Milky Way) when the sun is below the earth and 
does not cast its light about all the stars.' 2 

4 The Milky Way is the reflection {ανάκλασα) of the light of the 
stars which are not shone upon by the sun.' 3 

As Tannery 4 and Gomperz 5 point out, this conjecture, however 
ingenious, could easily have been disproved by simple observation. 
For Anaxagoras might have observed the obvious fact, noted as an 
objection by Aristotle, 6 that the Milky Way always retains the 
same position relatively to the fixed stars, whereas the hypothesis 
would require the trace of it to change its position along with the 
sun ; indeed the Milky Way should have coincided with the ecliptic, 
whereas it is actually inclined to it. Again, if the theory were true, 
an eclipse of the moon would have been bound to occur whenever the 
moon passed over the Milky Way, and it would have been easy to 
verify that this is not so. As the Milky Way is much longer than 
it is broad, it would seem that Anaxagoras thought that the flat 
earth was not round but 'elongated' (προμήκης), as Democritus 
afterwards conceived it to be, 7 though Democritus only made its 
its length half as much again as its breadth. 8 

Aristotle 9 adds an interesting criticism of this theory : ' Besides, 
if what is proved in the theorems on astronomy is correct, and the 
size of the sun is greater than that of the earth, and the distance of 
the stars from the earth is many times greater than the distance of 
the sun, just as the distance of the sun is many times greater than 
the distance of the moon, the cone emanating from the sun and 
marking the convergence of the rays would have its vertex not 
very far from the earth, and consequently the shadow of the earth, 

1 Arist. Meteorologica i. 8, 345 a 25-31 (Vors. i 2 , p. 308. 26-31). 

2 Aet. iii. 1. 5 {D. G. p. 365 ; Vors. i 2 , p. 308. 31). 

3 Hippol. Refut. i. 8. 10 {D. G. p. 561 ; Vors. i 2 , p. 302. 5); Diog. L. ii. 9 {Vors. 
\\ p. 294. 5). 

4 Tannery, op. cit., p. 279. 

5 Gomperz, Griechische Denker, i 3 , p. 179. 
β Aristotle, Meteorologica i. 8, 345 a 32. 

7 Eustathius in Homer. 11. vii. 446, p. 690 {Vors. i 2 , p. 367. 42). 

8 Agathemerus, i. 2 ( Vors. i 2 , p. 393. 10). 

9 Aristotle, Meteorologica i. 8, 345 b 1-9. 



1 Proclus in Titnaeum, p. 258 c (on Timaeus 38 d) ; Vors. i 2 , p. 308. 1-4. 

2 Ptolemy, Syntaxis ix. 1, vol. ii, p. 207, ed. Heiberg. 

3 Tannery, op. cit., p. 261. 

4 Aet. ii. 1.2 (D. G. p. 327 ; Vors. i 2 , p. 305. 44). 

5 Burnet, Early Greek Philosophy ', pp. 312, 313. 

6 Fr. 4 (Vors. i 2 , p. 315. 8-16). 



/ 



ch. x ANAXAGORAS 85 

which we call night, would not reach the stars at all. In fact the 
sun must embrace in his view all the stars and the earth cannot hide 
any one of them from him.' 

According to Proclus, 1 who quotes the authority of Eudemus, 
Anaxagoras anticipated Plato in holding that in the order of the 
revolution of the sun, moon, and planets round the earth the sun 
came next to the moon, whereas Ptolemy 2 says that according 
to * the more ancient ' astronomers (by which phrase he appears to 
mean the Chaldaeans 3 ) the order (starting from the earth) was 
Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn. 

'It seems clear that Anaxagoras held that there were other worlds 
than ours. Aetius, 4 it is true, includes Anaxagoras among those 
who said that there was only one world ; but the fragments must 
be held to be more authoritative, and one of these leaves no room 
for doubt on the subject. 6 According to this fragment 

1 Men were formed and the other animals which have life ; the 
men too have inhabited cities and cultivated fields as with us ; they 
have also a sun and moon and the rest as with us, and their earth 
produces for them many things of various kinds, the best of which 
they gather together into their dwellings and live upon.' 

Thus much have I said about separating ofT, to show that it will 
not be only with us that things are separated off, but elsewhere 
as well.' 6 



XI 

EMPEDOCLES 

The facts enabling the date of Empedocles of Agrigentum to be 
approximately determined are mainly given by Diogenes Laertius. 1 
His grandfather, also called Empedocles, won a victory in the 
horse-race at Olympia in 496/5 B. C. ; and Apollodorus said that 
his father was Meton, and that Empedocles himself went to Thurii 
shortly after its foundation. Thurii was founded in 445 B. c. and, 
when Diogenes Laertius says that Empedocles flourished in 01. 84 
(444/1), it is clear that the visit to Thurii was the basis for this 
assumption. According to Aristotle 2 he died at the age of sixty ; 
hence, assuming him to be forty in 444 B. c, we should have 484- 
424 B.C. as the date. But there is no reason why he should be 
assumed to have been just forty at the date of his visit to Thurii ; and 
other facts suggest that the date so arrived at is about ten years too 
late. Theophrastus said that Empedocles was born ' not long after 
Anaxagoras' ; 3 according to Alcidamas he and Zeno were pupils 
of Parmenides at the same time ; 4 and Satyrus said that Gorgias 
was a disciple of Empedocles. 5 Now Gorgias was a little older 
than Antiphon (of Rhamnus), who was born in 480 B.C. 6 It 
follows that we must go back at least to 490 B. C. for the birth of 
Empedocles ; most probably he lived from about 494 to 434 B.C. 7 

Empedocles is said to have been the inventor of rhetoric ; 8 as 
an active politician of democratic views he seems to have played 

1 Diog. L. viii. 51-74 (Vors. i 2 , pp. 149-53). 

2 In Diog. L. viii. 52 (Vors. i 2 , p. 150. 15). 

3 Theophrastus in Simpl. Phys. p. 25. 19 (D. G. p. 477 ; Vors. i 2 , p. 154. ■$?>). 

4 Diog. L. viii. 56 ( Vors. i 2 , p. 150. 41). 

5 Diog. L. viii. 58 ( Vors. i 2 , p. 151. 10). 

6 [Plutarch] Vit. X orat. i. 1. 9, p. 832 F (Vors. ii 2 . I, p. 546. 25). 

7 Cf. Diels' ' Empedokles und Gorgias', 2 (Berl. Sitztmgsb., 1884) ; Burnet, 
Early Greek Philosophy, pp. 228-9. 

8 Aristotle in Diog. L. viii. 57 (Vors. i 2 , p. 150. 46). 



EMPEDOCLES 87 

a prominent part in many a stirring incident ; he was a religious 
teacher, a physiologist and, according to Galen, 1 the founder of an 
Italian school of Medicine, which vied with those of Cos and 
Cnidus. That he was no mean poet is sufficiently attested by the 
fragments which survive, amounting to 350 (or so) lines or parts 
of lines in the case of the poem On Nature and over 100 in the 
case of the Purifications. 

Empedocles followed Anaximenes in holding that the heaven is 
a crystal sphere and that the fixed stars are attached to it. 2 The 
sphere, which is ' solid and made of air condensed or congealed 
by the action of fire, like crystal', 3 is, however, not quite spherical, 
the height from the earth to the heaven being less than its distance 
from it laterally, and the universe being thus shaped like an egg. 41 
While the fixed stars are attached to the crystal sphere, the planets 
are free. 5 

The sun's course is round the extreme circumference of the world 
(literally ' is the circuit of the limit of the world ') ; 6 in this particular 
Empedocles follows Anaximander. The circuit must be just inside 
the circumference because, under the heading ' tropics ' or ' turnings ' 
of the sun, Aetius says that, according to Empedocles, the sun is 
prevented from moving always in a straight line by the resistance 
of the enveloping sphere and by the tropic circles. 7 

A special feature of Empedocles' system is his explanation (1) of 
day and night, (2) of the nature of the sun. 

(1) Within the crystal sphere, and filling it, is a sphere consisting 
of two hemispheres, one of which is wholly of fire and therefore 
light, while the other is a mixture of air with a little fire, which 
mixture is darkness or night. The revolution of these two hemi- 

1 Galen, Meth. Med. i. I {Vors. i 2 , p. 154. 19-23). 

2 Aet. ii. 13. 11 {D. G. p. 342 ; Vors. i 2 , p. 162. 12). 

3 Aet. ii. 11. 2 (£>. G. p. 339 ; Vors. i 2 , p. 161. 40). 

4 Aet. ii. 31. 4 (D. G. p. 363 ; Vors. i 2 , p. 161. 34). The statement as to 
height and breadth is mathematically inconsistent with the comparison of the 
figure to an egg, unless we suppose that Empedocles regarded the section of it 
by the plane containing the surface of the earth as an oval and not a circle, 
which does not seem likely. If the said section is a circle, the figure would be 
what we call an oblate spheroid (the solid described by the revolution of an 
ellipse about its 7ninor axis) rather than egg-shaped. 

5 Aet. ii. 13. 11 (see above). 

6 Aet. ii. 1.4 {D. G. p. 328 ; Vors. i 2 , p. 161. 37) τον του ηλίου π^ρ'ώρομον uvat 
περιγραφην του πέρατος του κόσμου. 

7 Aet. ii. 23. 3 \ D - G - Ρ• 353 1 Vors. i 2 , p. 162. 37). 



88 EMPEDOCLES parti 

spheres round the earth produces at each point on its surface the 
succession of day and night. 1 The beginning of this motion was 
due to the collection of the mass of fire in one of the hemispheres, 
the result being that the pressure of the fire upset the equilibrium 
of the heaven and caused it to revolve. 2 Apparently connected 
with this theory of the two hemispheres is Empedocles' explanation 
of the difference between winter and summer. It is winter when 
the air (forming one hemisphere) gets the upper hand through 
condensation and is forced upwards (into the fiery hemisphere), and 
summer when the fire gets the upper hand and is forced downwards 
(into the dark hemisphere) ; 3 that is, in the winter the fire occupies 
less than half of the whole sphere of heaven, while in the summer 
it occupies more than half. The idea seems to be that the greater 
half of the sphere takes longer to revolve about a particular point 
on the earth's surface than the smaller half, and that this explains 
why the days are longer in the summer than in the winter. We 
are not told what was the axis about which the two hemispheres 
were supposed to revolve, but it seems hardly likely that Empedocles 
could have assumed a definite axis different from that of the daily 
rotation of the heavenly sphere. 

According to Empedocles it was the swiftness of the revolution 
of the heaven which kept the earth in its place, just as we may 
swing a cup with water in it round and round so that in some 
positions the top of the cup may actually be turned downwards 
without the water escaping. 4 The analogy is, of course, not a 
good one, because the water in that case is kept in its place by 
centrifugal force which throws it, as it were, against the side of the 
vessel, whereas the earth is presumably at rest in the centre during 
the revolution of the heaven, and is not acted on by such a force. 

Empedocles further held that the revolution of the heaven, which 
now takes 24 hours to complete, was formerly much slower. 
At one time a single revolution was only accomplished in a period 
equal to ten of our months ; later it required a period equal to seven 

1 Ps. Plut. Stromal, apud Euseb. Praep. Evang. i. 8. 10 (from Theophrastus) ; 
D. G. p. 582 ; Vors. i 2 , p. 158. 19-23. 

2 Ps. Plut. Stromal, loc. cit. {Vors. i 2 , p. 158. 33-4). 

3 Aet. iii. 8. 1 (D. G. p. 375 ; Vors. i 2 , p. 163. 16). 

4 Aristotle, De caelo ii. 13, 295 a 17 {Vors. r, p. 163. 39). 



CH. χι EMPEDOCLES 89 

of our months. 1 These views have, however, no astronomical basis ; 
they were put forward solely in order to explain the exceptions 
to the usual period of gestation afforded by ten-months' and seven- 
months' children, the period being in each case taken as one day ! 

Coming now to Empedocles's conception of the nature of the sun, 
we find the following opinions attributed to him : 

* The sun is, in its nature, not fire, but a reflection of fire similar 
to that which takes place from (the surface of) water.' 2 

1 There are two suns ; one is the original sun which is the fire 
in one hemisphere of the world, filling the whole hemisphere and 
always placed directly opposite the reflection of itself; the other 
is the apparent sun which is a reflection in the other hemisphere 
filled with air and an admixture of fire, and in this reflection what 
happens is that the light is bent back from the earth, which is 
circular, and is concentrated into the crystalline sun where it is 
carried round by the motion of the fiery (hemisphere). Or, to 
state the fact shortly, the sun is a reflection of the fire about the 
earth.' 3 

' The sun which consists of the reflection is equal in size to 
the earth.' 4 

* You laugh at Empedocles for saying that the sun is produced 
about the earth by a reflection of the light in the heaven and <4 once 
more flashes back to Olympus with fearless countenance ".' 5 

The second of the above passages is scarcely intelligible at the 
point where the reflection is called 'a reflection in the other 
hemisphere* \ it can hardly be in the other hemisphere because 
that hemisphere is night. Accordingly Tannery conjectures that the 
reading should be 'a reflection (invisible) in the other hemisphere'. 6 
The meaning must apparently be that the fire in the fiery hemi- 
sphere is reflected from the earth upon the crystal vault, the 
reflected rays being concentrated in what we see as the sun. The 
equality of the size of the sun and the earth may have been a hasty 
inference founded upon the supposition of an analogy with the 
recently discovered fact that the moon shines with light borrowed 

1 Aet. v. 18. I [D. G. p. 427 ; Vors. i 2 , p. 165. 31). 

2 Ps. Plut. Stroinat., loc. cit. {D. 67. p. 582 ; Vors. i 2 , p. 158. 35). 

3 Aet. ii. 20. 13 {D. G. p. 350; Vors. i 2 , p. 162. 18-24]. 

4 Aet. ii. 21. 2 (D. G. p. 351 ; Vors. i 2 , p. 162. 25). 

5 Plutarch, De Pyth. or. 12, p. 400 Β (Vors. i 2 , p. 188. 8-1 1). 
• Tannery, op. cit., p. 323. 



90 EMPEDOCLES part ι 

from the sun. 1 The theory that the sun which we see is a concen- 
tration of rays reflected from the earth upon the crystal sphere 
agrees exactly with the statement already quoted that the sun's 
course is confined just within the inner surface of the spherical 
envelope. Why it is just confined within the tropical circles and 
prevented from deviating further in latitude is not so clear. If, as 
Dreyer supposes, 2 the airy and the fiery hemispheres, which in turn 
occupy more than half of the heavenly sphere, ' thereby make 
the sun, the image of the fiery hemisphere, move south or north 
according to the seasons \ it would seem necessary to suppose that 
the advance of the hemisphere of fire in the summer (and its retreat 
in the winter) does not take place uniformly over the whole of its 
circular base (which is the division between the two hemispheres), 
i. e. in such a way that the base of the new hemisphere is parallel 
to the base of the old, but that the advance (or retreat) takes place 
obliquely with reference to the circular base, being greatest at a 
certain point on the rim of that base and least at the opposite 
point, so that the plane base of the new hemisphere is obliquely 
inclined to that of the old ; in other words, that the axis of the 
fiery hemisphere changes its position as the advance (or retreat) 
proceeds, and in fact swings gradually (completing an oscillation 
in a year) between two extreme positions inclined to the mean 
position at an angle equal to the obliquity of the ecliptic. But it is 
very unlikely that Empedocles, with his elementary notions of 
astronomy, worked out his theory in this way. 

It would appear that Empedocles' theory of the sun gave a lead 
to the later Pythagoreans, for we shall find Philolaus saying that 
1 there are in a manner two suns . . . unless [in Aetius's words] 

1 Cf. Plutarch, De fac. in orbe lunae 16, p. 929 Ε (Vors. i 2 , p. 187. 21-6): 
' There remains then the view of Empedocles that the illumination which we 
get here from the moon is produced by a sort of reflection of the sun at the 
moon [the same word άνάκΚιισιν being used in this case]. Hence we get neither 
heat nor brightness from it, whereas we should expect both if there had been 
a kindling and mixing of (the) lights, and, just as when sounds are reflected 
the echo is less distinct than the original sound, . . . . " even so the ray which 
struck the moon's wide orb" passes on to us a reflux which is weak and indistinct, 
owing to the loss of power due to the reflection.' But, if Empedocles spoke in 
this way of the moon's light, he could hardly have conceived the light of the 
sun, which is bright and hot, to be a reflection of light in the same sense as the 
light of the moon is; the 'reflection of light' which constitutes the sun is more 
like the effect of a burning-glass than ordinary reflection. 

2 Dreyer, Planetary Systems, p. 25. 



CH.XI EMPEDOCLES 91 

we prefer to say that there are three, the third consisting of the 
rays which are reflected again from the mirror or lens [the second 
sun] and spread in our direction '- 1 

Empedocles does not seem to have mentioned the annual motion 
of the sun relatively to the fixed stars, although, as we have seen, 
he speaks of the tropic circles as limiting its motion (i. e. motion in 
latitude). 

Empedocles, like Anaxagoras, held that the moon shone with 
light borrowed from the sun. 2 The moon itself he regarded as 
' a mass of frozen air, like hail, surrounded by the sphere of 
fire ', 3 or as ' condensed air, cloudlike, solidified (or congealed) 
by fire, so that it is of mixed composition \ 4 This idea may have 
been put forward to account for the apparent change of shape in 
the phases ; for we find Plutarch saying that ' the apparent form 
of the moon, when the month is half past, is not spherical, but 
lentil-shaped and like a disc, and, in the opinion of Empedocles, its 
actual substance is so too \ 5 

The stars he thought to be 'of fire (arising) out of the fiery 
(element) which the air contained in itself but squeezed out upwards 
in the original separation \ 6 

We are not definitely told whether Empedocles held the earth 
to be spherical or flat. He might, it is true, have adopted the view 
of the Pythagorean school and Parmenides that it was spherical, 
but it is more probable that he considered it to be flat. For we 
are told that he regarded the moon as * like a disc ' 7 ; in this he 
probably followed Anaxagoras, who undoubtedly thought the earth 
flat, and therefore most probably the moon also. 

He also shared the view of Anaxagoras that the axis of the 
world was originally perpendicular to the surface of the earth, the 
north pole being in the zenith, and that it was displaced afterwards. 
This view Anaxagoras combined with the hypothesis of a flat 

1 Aet. ii. 20. 12 (D. G. p. 349 ; Vors. i 2 , p. 237. 39). 

2 Fr. 43, quoted in note on preceding page ; Aet. ii. 28. 5 (D. G. p. 358 ; 
Vors. i 2 , p. 162. 48). 

3 Plutarch, Defac. in orbe lunae 5, p. 922 C ( Vors. i 2 , p. 162. 43). 

4 Aet. ii. 25. 15 (D. G. p. 357 ; Vors. i 2 , p. 162. 41). 

5 Plutarch, Quaest. Rom. 101, p. 288 Β {Vors. i 2 , p. 162. 45). 

6 Aet. ii. 13. 2 (D. G. p. 341 ; Vors. i 2 , p. 162. 10). 

7 Diog. L. viii. 77 (Vors. i 2 , p. 153. 37) ; Aet. ii. 27. 3 (D.G. p. 358; Vors. i 2 , 
p. 162. 44). 



9a EMPEDOCLES part ι 

earth ; indeed, a flat earth is almost necessary if the axis of the 
universe was originally perpendicular to its surface. Empedocles, 
however, differed from Anaxagoras in his explanation of the cause 
of the subsequent displacement ; whereas Anaxagoras could only 
account for it tentatively by assuming 'design', Empedocles gave 
a mechanical explanation : 

1 The air having yielded to the force of the sun, the north pole 
became inclined, the northern parts were heightened, and the 
southern lowered, and the whole universe was thereby affected.' * 

* There are many fires burning beneath the earth ', 2 said 
Empedocles. He seems to have inferred this truth from the 
existence of hot springs, the water of which he supposed to be 
heated, like the water in baths, by running a long course, as it 
were in tubes, through fire. 3 

According to Empedocles the sun is a great collection of fire and 
greater than the moon, 4 and the sun is twice as distant from the 
earth as the moon is. 5 

He was aware of the true explanation of eclipses of the sun, for 
he says that 

1 The moon shuts off the beams of the sun as it passes across it, 
and darkens so much of the earth as the breadth of the blue-eyed 
moon amounts to.' 6 

With this may be compared his description of night as caused 
by the shadow of the earth which obstructs the rays of the sun 
as the sun passes under the earth. 7 

Empedocles' one important scientific achievement, so far as we 
know, was his theory that light travels and takes time to pass from 
one point to another. The theory is alluded to by Aristotle in 
the following passages : 

1 Aet. ii. 8. 2 (D. G. p. 338 ; Vors. i 2 , p. 162. 35). 

2 Fr. 52 (Vors. i 2 , p. 189. 14). 

3 Seneca, Ναι. Quaest. iii. 24, quoted by Burnet, Early Greek Philosophy ', 
p. 277, note. 

4 Diog. L. viii. yj {Vors. i 2 , p. 153. 36). 

5 Aet. ii. 31. 1 (D. G. p. 362 ; Vors. i 2 , p. 163. 1-3). I follow the text as 
corrected by Diels after Karsten. The reading of Stobaeus is corrupt. That of 
the Placita says that the moon is twice as far from the sun as it is from the 
earth. 

6 Fr. 42 (Vors. i 2 , p. 187. 28) : cf. Aet. ii. 24. 7 (D. G. p. 354; Vors. i 2 , 
p. 162. 40). 

7 Fr. 48 (Vors. i 2 , p. 188.31). 



ch.xi EMPEDOCLES 93 

1 Empedocles, for instance, says that the light from the sun 
reaches the intervening space before it reaches the eye or the earth. 
And this might well seem to be the fact. For, when a thing is 
moved, it is moved from one place to another, and hence a certain 
time must elapse during which it is being moved from the one 
place to the other. But every period is divisible. Therefore there 
was a time when the ray was not yet seen, but was being trans- 
mitted through the medium.' * 

1 Empedocles represented light as moving in space and arriving 
at a given point of time between the earth and that which surrounds 
it, without our perceiving its motion.' 2 

Aristotle of course rejected this theory because he himself held 
a different view, namely, that light was not a movement in space 
but was a qualitative change of the transparent medium which, he 
considered, could be changed all at once and not only (say) half 
at a time, just as a mass of water is all simultaneously congealed. 3 
But he had no better argument to oppose to Empedocles than that 
'though a movement of light might elude our observation within 
a short distance, that it should do so all the way from east to west 
is too much to assume \* 

1 Aristotle, De sensu 6, 446 a 25-b 2. 

2 Aristotle, De anima ii. 7. 418 b2i. 

3 Aristotle, De sensu 6, 447 a 1-3. 

4 Aristotle, De anima ii. 7, 418 b 24. 



nHVp^MMBMMMNHHMMBHai^na^v•••»• * m ^wer , ™ > "^»" ,, ^ ,i ^r™^Wi 



XII 

THE PYTHAGOREANS 

In a former chapter we tried to differentiate from the astronomical 
system of ' the Pythagoreans ' the views put forward by the Master 
himself, and we saw reason for believing that he was the first to give 
spherical shape to the earth and the heavenly bodies generally, 
and to assign to the planets a revolution of their own in a sense 
opposite to that of the daily rotation of the sphere of the fixed 
stars about the earth as centre. 

But a much more remarkable development was to follow in the 
Pythagorean school. This was nothing less than the abandonment 
of the geocentric hypothesis, and the reduction of the earth to the 
status of a planet like the others. Aetius (probably on the authority 
of Theophrastus) attributes the resulting system to Philolaus, 
Aristotle to ' the Pythagoreans '. 

Schiaparelli J sets out the considerations which may have sug- 
gested to the Pythagoreans the necessity of setting the earth itself 
in motion. If the proper movement of the sun, moon, and planets 
along the zodiac had been a rotation about the same axis as that 
of the daily rotation of the fixed stars, it would have been easy 
to account for the special movements of the former heavenly bodies 
by assuming for each of them a daily rotation somewhat slower 
than that of the fixed stars ; if the movement of each of them 
had been thus simple, a moving force at the centre operating 
with various degrees of intensity (depending on distance and the 
numerical laws of harmony) would have served to explain every- 
thing. But, since the daily rotation follows the plane of the 
equator, and while special movement of the planets follows the 
plane of the ecliptic, it is clear that, with one single moving force 

1 Schiaparelli, I firecursori di Copernico nelV antichita (Milano, Hoepli, 
1873)» P• 4. 



THE PYTHAGOREANS 95 

situated at the centre, it was not possible to account for both 
movements. Hence the necessity of attributing the daily rotation, 
which is apparently common to the fixed stars and the planets, 
to a motion of the earth itself. But another reason too would 
compel the Pythagoreans to avoid attributing to the sun, moon, 
and planets the movement compounded of the daily rotation and 
the special movement along the zodiac. For such a composite 
movement would take place in a direction and with a velocity 
continually altering and it would follow that, if at a given instant 
the harmonical proportions of the velocities and the distances held 
good, these proportions would not hold good for the next instant. 
Accordingly it was necessary to assign to each heavenly body one 
single simple and uniform movement, and this could not be 
realized except by attributing to the earth that one of the compo- 
nent movements which observation showed to be common to all 
the stars. 

Whether the system attributed to Philolaus was really founded 
on arguments so scientific, combining the data furnished by observa- 
tions with an antecedent principle based on the nature of things 
and on a living spirit animating the world, must be left an open 
question. 

It is time to attempt a description of the system itself, and 
I think that this can best be done in the words of our authorities. 



• 



Motion round the central fire. 



' While most philosophers say that the earth lies in the centre . . . 
the philosophers of Italy, the so-called Pythagoreans, assert the 
tu contrary. They say that there is fire in the middle, and the earth, 
being one of the stars, is carried round the centre, and so produces 
night and day. \ They also assume another earth opposite to ours, 
which they call counter-earth, and in this they are not seeking explana- 
tions and causes to fit the observed phenomena, but they are rather 
trying to force the phenomena into agreement with explanations 
and views of their own and so adjust things. Many others might 
agree with them that the place in the centre should not be assigned 
to the earth, if they looked for the truth not in the observed 
facts but in a priori arguments. For they consider that the 
worthiest place is appropriate to the worthiest occupant, and fire 
is worthier than earth, the limit worthier than the intervening parts, 
while the extremity and the centre are limits ; arguing from these 
considerations they think that it is not the earth which is in the 



9 6 THE PYTHAGOREANS parti 

centre of the (heavenly) sphere but rather the fire. Further, the 
Pythagoreans give the additional reason that it is most fitting that 
the most important part of the All— and the centre may be so 
described— should be safe-guarded ; they accordingly give the name 
of'Zeus's watch-tower" to the fire which occupies this position, 
the term " centre " being here used absolutely and the implication 
being that the centre of the (thing as a) magnitude is also the centre 
of the thing in its nature .... Such are the opinions of certain 
philosophers about the position of the earth ; and their opinions 
about its rest or motion correspond. For they do not all take 
the same view ; those who say that the earth does not so much 
as occupy the centre make it revolve in a circle round that 
centre, and not only the earth but the counter-earth also, as we 
said before. Some again think that there may be even more 
bodies of the kind revolving round the centre ; only they are 
invisible to us because of the interposition of the earth. This they 
give as the reason why there are more eclipses of the moon than of 
the sun ; for the moon is obscured by each of the other revolving 
bodies as well, and not only by the earth. The fact that the earth 
is not the centre, but is at a distance represented by the whole 
(depth, i.e. radius) of half the sphere (in which it revolves) con- 
stitutes, in their opinion,, no reason why the phenomena should not 
present the same appearance to us if we lived (on an earth) away 
from the centre as they would if our earth were at the centre ; 
seeing that, as it is, we are at a distance (from the centre) repre- 
sented by half the earth's diameter and yet this does not make any 
obvious difference.' * 

' The Pythagoreans, on the other hand, say that the earth is not 
at the centre, but that in the centre of the universe is fire, while 
round the centre revolves the counter-earth, itself an earth, and 
called counter-earth because it is opposite to our earth, and next 
to the counter-earth comes our earth, which itself also revolves 
round the centre, and next to the earth the moon ; this is stated 
by Aristotle in his work on the Pythagoreans. The earth then, 
being like one of the stars, moves round the centre and, according 
to its position with reference to the sun, makes night and day. The 
counter-earth, as it moves round the centre and accompanies our 
earth, is invisible to us because the body of the earth is continually 
interposed in our way. . . . The more genuine exponents of the 
doctrine describe as fire at the centre the creative force which from 
the centre imparts life to all the earth and warms afresh the part of 
it which has cooled. Hence some call this fire the Tower of Zeus, 
as Aristotle states in his Pythagorean Philosophy, others the 

1 Aristotle, De caelo ii. 13, 293 a 18-b 30 (partly quoted in Vors. i 2 , p. 278. 
4-20, 38-40). 



ch.xii THE PYTHAGOREANS 97 

Watch-tower of Zeus, as Aristotle calls it here [Be caelo ii. 13], 
and others again the Throne of Zeus, if we may credit different 
authorities. L-They called the earth a star as being itself too 
an instrument of timeTjfor it is the cause of days and nights, 
since it makes day w"nen it is lit up in that part of it which 
faces the sun, and it makes night throughout the cone formed by 
its shadow.' * 

* Philolaus calls the fire in the middle about the centre the Hearth 
of the universe^ the House of Zeus, the Mother of the Gods, the 
Altar, Bond and Measure of Nature. And again he assumes another 
fire in the uppermost place, the fire which encloses (all). Now the 
middle is naturally first in order, and round it ten divine bodies 
move as in a dance, [the heaven] and (after the sphere of the fixed 
stars) 2 the five planets, after them the sun, under it the moon, 
under the moon the earth, and under the earth the counter-earth ; 
after all these comes the fire which is placed like a hearth round 
the centre. The uppermost part of the (fire) which encloses (all), 
in which the elements exist in all their purity, he calls Olympus, 
and the parts under the moving Olympus, where are ranged the 
five planets with the moon and the sun, he calls the Universe, and 
lastly the part below these, the part below the moon and round 
the earth, where are the things which suffer change and becoming, 
he calls the Heaven.' 3 

4 Philolaus the Pythagorean places the fire in the middle (for this 
is the Hearth of the All), second to it he puts the counter-earth, 
and third the inhabited earth which is placed opposite to, and 
revolves with, the counter-earth ; this is the reason why those who 
live in the counter-earth are invisible to those who live in our 
earth/ 4 

' Thejrovejxdng^ 
and the Creating God established it there as a sort of keel to the 
(sphere) of the All/ 5 

~ ( Others maintain that the earth remains at rest. But Philolaus 
the Pythagorean held that it revolves round the fire in an oblique 
circle in the same way as the sun and moon/ 6 ' * 



1 Simplicius on De caelo ii. 13, 293 a 15, pp. 511. 25-34 and 512. 9-17 Heib. 
( Vors. i 2 , p. 278. 20-36). 

2 The words are supplied by Diels in view of similar words in a passage of 
Alexander Aphrodisiensis quoted below (Alex, on Metaph. 985 b 26, p. 540 b 4-7 
Brandis, p. 38. 22-39. 3 Hayduck). 

3 Aet. ii. 7. 7 {D. G. p. 336-7 ; Vors. i 2 , p. 237. 13 sqq.). This and the next 
extract probably come from Theophrastus, through Posidonius. 

4 Aet. iii. II. 3 {D. G. p. 377 ; Vors. i 2 , p. 237. 27 sq.). 
6 Aet. ii. 4. 15 [D.G. p. 332 ; Vors. i 2 , p. 237. 31). 

6 Aet. iii. 13. 1, 2 (D. G. p. 378 ; Vors. i 2 , p. 237. 46). 

1410 Η 



9 8 THE PYTHAGOREANS parti 

As regards the assumption of ten bodies we have the following 
further explanations. In a passage of the Metaphysics Aristotle 
is describing how the Pythagoreans find the elements of all existing 
things in numbers ; he then proceeds thus : 

' They conceived that the whole heaven is harmony and number ; 
thus, whatever admitted facts they were in a position to prove in 
the domain of numbers and harmonies, they put these together 
and adapted them to the properties and parts of the heaven and 
its whole arrangement. And if there was anything wanting any- 
where, they left no stone unturned to make their whole system 
coherent. For example, regarding as they do the number ten as 
perfect and as embracing the whole nature of numbers, they say 
that the bodies moving in the heaven are also ten in number, and, 
as those which we see are only nine, they make the counter-earth 
a tenth.' * 

Alexander adds in his note on this passage : 

1 If any of the phenomena of the heaven showed any disagree- 
ment with the sequence in numbers, they made the necessary 
addition themselves, and tried to fill up any gap, in order to make 
their system as a whole agree with the numbers. Thus, considering 
the number ten to be a perfect number, and seeing the number 
of the moving spheres shown by observation to be nine only, those 
of the planets being seven, that of the fixed stars an eighth, and 
the earth a ninth (for they considered that the earth too moved 
in a circle about the Hearth which remains fixed and, in their view, 
is fire), they straightway added to them in their doctrine the 
counter-earth as well, which they supposed to move counter to 
the earth and so to be invisible to the inhabitants of the earth.' 2 

Speaking of the sun in an earlier passage, Alexander says : 

1 (The sun) they placed seventh in order among the ten bodies 
which move about the centre, the Hearth ; for the movement of the 
sun comes next after (that of) the sphere of the fixed stars and 
the five movements belonging to the planets, while after the sun 
the moon comes eighth, and the earth ninth, after which again comes 
the counter-earth.' 3 

The system may be described briefly thus. The universe is 
spherical in shape and finite in size. Outside it is infinite void 

1 Aristotle, Metaph. A. 5, 986 a 2-12 (Vors. i 3 , p. 270. 40-47). 

2 Alexander on Metaph. 986 a 3 (p. 542 a 35-b 5 Brandis, p. 40, 24-41. 1 
Η ay duck). 

8 Ibid. 985 b 26 (p. 540 b 4-7 Brandis, p. 38. 22-39. 3 Hayduck). 



ch.xii THE PYTHAGOREANS 99 

which enables the universe to breathe, as it were. At the centre is 
the central fire, the Hearth of the universe, called by the various 
names, the Tower or Watch-tower of Zeus, the Throne of Zeus, 
the House of Zeus, the Mother of the Gods, the Altar, Bond and 
Measure of Nature. In this central fire is located the governing 
principle, the force which directs the movement and activity of 
the universe. The outside boundary of the sphere is an envelope 
of fire ; this is called Olympus, and in this region the elements 
are found in all their purity ; below this is the Universe. In the 
universe there revolve in circles round the central fire the following 
bodies. Nearest to the central fire revolves the counter-earth, 
which always accompanies the earth, the orbit of the earth coming 
next to that of the counter-earth ; next to the earth, reckoning 
in order from the centre outwards, comes the moon, next to the 
moon the sun, next to the sun the five planets, and last of all, 
outside the orbits of the planets, the sphere of the fixed stars. 

The counter-earth, which accompanies the earth and revolves 
in a smaller orbit, is not seen by us because the hemisphere of the 
earth on which we live is turned away from the counter-earth. 
It follows that our hemisphere is always turned away from the 
central fire, that is, it faces outwards from the orbit towards 
Olympus (the analogy of the moon which always turns one side 
towards us may have suggested this) ; this involves a rotation of 
the earth about its axis completed in the same time as it takes the 
earth to complete a revolution about the central fire. 

What was the object of introducing the counter-earth which 

we never see? Aristotle says in one place that it was to bring 

up the number of the moving bodies to ten, the perfect number 

according to the Pythagoreans. But clearly Aristotle knew better ; 

indeed he himself indicates the true reason in another passage 

where he says that eclipses of the moon were considered to be 

due sometimes to the interposition of the earth, sometimes to the 

interposition of the counter-earth (to say nothing of other bodies 

of the same sort assumed by ' some ' in order to explain why there 

appear to be more lunar eclipses than solar). 1 The counter-earth, 

1 De caelo ii. 13, 293 b 21. Cf. Aet. ii. 29. 4 (D. G. p. 360 ; Vors. i 2 , p. 277. 46) : 
* Some of the Pythagoreans, according to the account of Aristotle and the 
statement of Philippus of Opus, say that the moon is eclipsed through reflection 
and the interposition sometimes of the earth, sometimes of the counter-earth.' 

Η 2 




1 ) 

J ' 

I > I 






ioo THE PYTHAGOREANS parti 

therefore, we may take to have been invented for the purpose of 
explaining eclipses of the moon, and particularly the frequency 
with which they occur. 

The earth revolves round the central fire in the same sense as the 
sun and moon (that is, from west to east), but its orbit is obliquely 
inclined ; that is to say, the earth moves in the plane of the equator, 
the sun and the moon in the plane of the zodiac circle. It would 
no doubt be in this way that Philolaus would explain the seasons. 

Next we are told that the revolution of the earth produces day 
and night, which depend on its position relatively to the sun ; 
it is day in that part which is lit up by the sun and night in the 
cone formed by the earth's shadow. As the same hemisphere is 
always turned outwards, it seems to follow from the natural 
meaning of these expressions that the earth completes one revolu- 
tion round the central fire in a day and a night, or in 24 hours. 1 
This would, of course, account for the apparent diurnal rotation 
of the heavens from east to west ; from this point of view it is 
equivalent to the rotation of the earth on its own axis in 24 hours 
But there is a considerable difficulty here, of which, if we may trust 
Aristotle, the Pythagoreans made light. According to him the 
Pythagoreans said that whether (1) the earth revolves in a circle 
round the centre of the universe or (2) the earth is itself stationary 
at that centre could make no difference in the appearance of the 
phenomena as observed by us. They argued that, even if we 
assume the earth to be at the centre, there is a distance between 
the centre and an observer on the earth's surface equal to the 
radius of the earth. On their assumption that the earth revolves 
round the centre of the universe, the distance of an observer from 
that centre would be greater than the radius of the earth's orbit ; 
therefore to assert that the phenomena under the two assumptions 
would be exactly the same was to argue in effect that parallax 
is as negligible in one case as in the other. This is a somewhat 
extreme case of making the phenomena fit a preconceived hypo- 
thesis ; but we may no doubt infer that the difficulty would lead 
the Pythagoreans to maintain that the distance of the earth from 
the centre of the universe was very small relatively to the distance 

1 Burnet apparently disputes this inference {Early Greek Philosophy, p. 352, 
note). We shall return to the point later. 



' , • • . ' • • • ' 



1 ι 



t t I 

> 



ch.xii THE PYTHAGOREANS ιοί 

of the other heavenly bodies from that centre, and that the radius 
of the earth's orbit was not in fact many times greater than the 
radius of the earth itself. 1 

But a still greater difficulty remains. On the assumption that 
the earth revolves round the central fire in a day and a night, and 
that the sun, the moon, and the five planets complete one revolution 
in their own several periods respectively, the observed movements 
of these heavenly bodies are accounted for. But, since the apparent 
daily rotation of the heavens is due to the revolution of the earth 
about the central fire in a day and a night, it would follow that the 
sphere of the fixed stars does not move at all, and therefore it 
could not be said that ' ten bodies ' (of which that sphere is one) 
revolve about the central fire. 

Boeckh suggested in his Philolaus that the motion of the sphere 
of the fixed stars could only be the precession of the equinoxes. 
This he thought might have been discovered by the Egyptians, 2 
and Lepsius, later, took the same view, even suggesting that the 
Egyptians might have communicated the discovery to Eudoxus. 3 
Boeckh afterwards, as a result of a study of Egyptian monuments, 
withdrew his suggestion ; 4 but, later still, he seems to have taken 
it up again as preferable to the supposition of a very slow movement 
serving no purpose and frankly faked. But, so far as we know, 
Hipparchus was the first to discover precession. Martin passed 
through two stages corresponding to Boeckh's first and second. 
In his commentary on the Timaeus of Plato, Martin observed 
that precession only ' required long and steady observations, with- 
out any mathematical theory, in order to be recognized '; 5 but 
Martin, too, changed his opinion later and satisfied himself that 
precession was not known to any of Hipparchus's predecessors. 6 

Schiaparelli thought it probable that Philolaus attributed no 

1 Schiaparelli, 1 firecursori, p. 6. 

2 Boeckh, Philolaos des Pythagoreers Lehren, 1819, pp. 11 8, 1 19. 

3 Lepsius, Chronologie der alien Aegyftter, p. 207. 

4 Boeckh, Manetho und die Hundsternperiode, 1845, p. 54. 

5 Martin, Etudes sur le Timee de Platon, ii, p. 98. 

6 Martin, ' La precession des equinoxes a-t-elle ete connue des Egyptiens ou 
de quelque autre peuple avant Hipparque?' in vol. viii, pt. 1, of Memoires de 
rAcademie des Inscriptions et Belles-lettres, Savants Ε t rangers, Paris, 1869 ; 
see also Hypothese astronomique de Philolaus ; by the same author, Rome, 1 872, 
p. 14. 



ίο» THE PYTHAGOREANS parti 

movement to the sphere of the fixed stars, 1 his ground being the 
following. Censorinus attributes to Philolaus the statements that 
a ' Great Year ' consists of 59 years, and that the solar year has 
364 \ days. This gives a Great Year consisting of 21,505! days, 
which period contains very approximately 2, revolutions of Saturn, 
5 of Jupiter, 31 of Mars, 59 of the sun, Mercury, and Venus, and 
729 of the moon. 2 If, then, says Schiaparelli, Philolaus had attributed 
any movement to the stars, he would probably have included its 
period in his Great Year ; which apparently he did not. Tannery, 
however, has given reason for thinking that the 729 lunations, and 
consequently the 364! days, were not the result of any independent 
calculation made by Philolaus, but were an arbitrary variation 
from the figures of Oenopides of Chios, of whom we are told by 
Censorinus that he made the year to be ^^ii days, so that 59 
years would give 21,557 days or 730 lunations, not 729. Philolaus 
said, as Plato said after him, that the cube of 9 represents the 
number of months in a Great Year, and so it does less 1 ; the 
arbitrary variation is characteristic of the Pythagorean fanciful 
speculations with regard to numbers. 3 

1 Schiaparelli, I precursor i, p. 7. 

2 Schiaparelli (/ precursori, p. 8, note) compares the periods of revolution 
based on the figures attributed to Philolaus with the true periods, thus : 

Period of revolution. 



Planet. Philolaus. Modern view. 

Saturn 1075275 days 10759.22 days 

Jupiter 4301-10 „ 4332-58 „ 

Mars 69371 „ ■ 686.98 „ 

Venus ^ 



Mercury \ 364.50 „ 365-26 „ 

Sun ) 

Moon 29-50 „ 29.53 „ 

Schiaparelli admits that the number of days for Mars (69371) is uncertain, 
as it is not clear that Philolaus assumed 31 revolutions of Mars in his Great 
Year. But neither does there appear to be any evidence that he definitely fixed 
the number of revolutions made by the other planets in the Great Year. 

■ Tannery, ' La grande annee d'Aristarque', in Memoires de la S octet e des 
sciences physiques et naturelles de Bordeaux, 3 Θ ser. iv, 1888, p. 90. 

Tannery holds that Philolaus simply took his Great Year, equal to 59 solar 
years, from Oenopides, while Oenopides, arrived at it in a very simple way, 
namely, by taking the number of days in the year as 365, and the period of the 
moon as 29^ days, and observing the natural inference that, in whole numbers, 
59 years are equal to 730 lunar months, after which he had only to determine 
the number of days in 730 lunar months. 



ch.xii THE PYTHAGOREANS 103 

But indeed, as Burnet points out, 1 it is incredible that the 
Pythagoreans should have put forward the theory that the sphere 
of the fixed stars is absolutely stationary. Such a suggestion 
would have seemed such a startling paradox that it is inconceivable 
that Aristotle should have said nothing about it, especially as he 
made the circular motion of the heavens the keystone of his own 
system. As it is, he does not attribute to any one the view that 
the heavens are stationary ; and, in writing of the Pythagorean 
system, he makes it perfectly clear that the bodies moving in the 
heaven are ten in number, 2 from which it follows that the sphere 
of the fixed stars (which is one of the ten) must move. It may 
be observed, too, that Alcmaeon, whom Aristotle mentions as 
having held views similar to the Pythagoreans, distinctly said that 
* all the divine bodies, the moon, the sun, the stars, and the whole 
heaven, move continually '. 3 

Now, if the Pythagoreans gave a movement of rotation to the 
sphere of the fixed stars, there are three possibilities. The first 
is that they may have assumed the universe as a whole to share 
in the rotation of the sphere of the fixed stars, while the independent 
revolutions of the earth, sun, moon, and planets were all in addition 
to their rotation as part of the universe. If this were the assumption, 
the rotation of the whole universe might be at any speed whatever 
without altering the phenomena as observed by us ; the phenomena 
would present exactly the same appearance to us as they would 
on the assumption that the sphere of the fixed stars is stationary, 
and the planets, sun, moon, earth, and counter-earth have only 
their own proper revolutions round the central fire ; only to a 
person situated at the central fire, supposed exempt from the 
general movement, would the general movement of the universe 
be perceptible. Thus the assumption of such a general movement 
would serve no purpose (apart from the objection that it would 
leave the speed of the rotation of the whole universe quite 
indeterminate) ; indeed, it would defeat what seems to have been 

1 Burnet, Early Greek Philosophy, p. 347. 

2 Aristotle, Metaph.h.. 5,986a ΙΟ τα φερόμενα κατά. τον ovpavbv SeVa μεν dvai φασιν. 
Cf. the passages of Alexander, quoted above (p. 98) ; also Simplicius on Decaelo 
2 93 a l S (Ρ• 5 12 • 5)» * They wished to bring up to ten the number of the bodies 
which have a circular motion (κυκλοφορητικών).' 

3 Aristotle, De anima i. 2, 405 a 33. 



I04 THE PYTHAGOREANS parti 

the whole object of Philolaus's scheme, namely, to separate the 
daily rotation from the periodical revolutions of the sun, moon, 
earth, and planets, and to account for all the phenomena by simple 
motions instead of a combination of two in each case. 

The second possibility is only slightly different. The sphere of 
the fixed stars might have a movement of rotation and carry with 
it all the heavenly bodies except the earth (and of course its 
inseparable companion, the counter-earth). The effect would be 
that the earth (with the counter-earth) would complete an actual 
revolution round the central fire in a period greater or less than 
24 hours according to the speed and the direction of the rotation 
of the rest of the heavenly bodies ; 1 the period would be less than 
24 hours if the latter rotation were in the same sense as that of 
the earth's revolution from west to east, and greater if it were in the 
opposite sense, from east to west. This alternative is more compli- 
cated than the first, and is open to the same or stronger objections. 

The third possibility is that the sun, moon, planets, earth, and 
counter-earth have their own special movements only, and that 
the sphere of the fixed stars moves very slowly ', so slowly that its 
movement is imperceptible. This is the view of Martin 2 and of 
A pelt, 3 and it amounts to assuming that Philolaus gave a move- 
ment to the sphere of the fixed stars which, though it is not the 
precession of the equinoxes, is something very like it. If this is 
right, we must suppose that Philolaus gave the sphere of the fixed 
stars a merely nominal rotation for the sake of uniformity and 
nothing else ; and perhaps, as Martin says, to assume an imperceptible 
motion would not be a greater difficulty for Philolaus than it was 
to postulate an invisible planet or to maintain that the enormous 
parallaxes which would be produced by the daily revolution of 
the earth about the central fire are negligible. 

It is to be feared that a convincing solution of the puzzle will 

1 Martin [Hypothhe astronomique de Philolaus ; Rome, 1872, p. 16) compares 
an allusion in Ptolemy's Syntaxis (1.7, p. 24. 11-13 Heib.) to the possibility 
of assuming (as an alternative to a scheme in which the fixed stars are stationary 
and the earth rotates 011 its own axis once in twenty-four hours) that both the 
earth and the sphere of the fixed stars rotate, at different speeds, about one and 
the same axis, the axis of the earth. 

2 Martin, Hypothlse astronomique de Philolaus^ pp. 14-16. 

3 Apelt, Untersuchungen iiber die Philosophic und Physik der Alten (Abh. 
der Fries' schen Schule s Heft I, p. 68 j. 



ch.xii THE PYTHAGOREANS 105 

never be found. After all that has been written on the subject, 
Gomperz 1 still seems to prefer Boeckh's original suggestion that the 
movement attributed by Philolaus to the fixed stars was actually 
the precession of the equinoxes, but the new matter contained in 
his note on the subject does not help his case. He relies partly 
on the a priori arguments originally put forward by Martin ; ' it 
is', he suggests, ' in itself hardly credible that a deviation in the 
position of the luminaries which in the course of a single year 
amounts to more than 50 seconds of an arc could remain unnoticed 
for long ' ; he is aware ; however, that Martin himself, as the result 
of further investigation, could find no confirmation of his earlier 
view. He admits, too, that the Babylonians were still unacquainted 
with precession in the third century B.C. 2 The other main argument 
used by Gomperz is that the estimates of the angular velocities 
of the planetary movements which go back to Philolaus or other 
early Pythagoreans are approximately correct, while only prolonged 
observations of the stars could have made them so. But, so far as 
Philolaus is concerned, the data are apparently the same as those 
from which Schiaparelli drew the opposite inference, namely, that 
Philolaus was not aware of precession and considered the sphere of 
the fixed stars to be stationary ! 

Harmony and distances. 

1 Philolaus holds that all things take place by necessity and by 
harmony.' 3 

* It is clear too from this that, when it is asserted that the move- 
ment of the stars produces harmony, the sounds which they make 
being in accord, the statement, although it is a brilliant and remark-, 
able suggestion on the part of its authors, does not represent the 
truth. I refer to the view of those who think it inevitable that, 
when bodies of such size move, they must produce a sound ; 
this, they argue, is observed even of bodies within our experience 
which neither possess equal mass nor move with the same speed ; 
hence, when the sun and moon, and the stars which are so many 
and of such size move with such a velocity, it is impossible that they 

1 Gomperz, Griechische Denker, i 8 , p. 93, and note on pp. 430, 431. 

2 Gomperz (p. 431) gives this as the opinion of the highest authority on the 
subject, Pater Kugler, who is to argue the point anew in a forthcoming tract, 
' Im Bannkreis Babels '. This must be set against the opposite inference drawn 
by Burnet (Early Greek Philosophy, p. 25, note) from another work of Kugler' s, 
and apparently confirmed by Hilprecht (The Babylonian Expedition of the 
University of Pennsylvania, Philadelphia, 1906). 

3 Diog. L. viii. 84 ( Vors. i 3 , p. 233. 33). 



ιοό THE PYTHAGOREANS parti 

should not produce a sound of intolerable loudness. Supposing 
then that this is the case, and that the velocities depending on 
their distances correspond to the ratios representing chords, they 
say that the tones produced by the stars moving in a circle are in 
harmony. But, as it must seem absurd that we should not all hear 
these tones, they say the reason of this is that the sound is already 
going on at the moment we are born, so that it is not distinguishable 
by contrast with its opposite, silence ; for the distinction between 
vocal sound and silence involves comparison between them ; thus a 
coppersmith is apparently indifferent to noise through being accus- 
tomed to it, and so it must be with men in general.' * 

1 For (they said that) the bodies which revolve round the centre 
have their distances in proportion, and some revolve more quickly, 
others more slowly, the sound which they make during this motion 
being deep in the case of the slower and high in the case of the 
quicker; these sounds, then, depending on the ratio of their distances, 
are such that their combined effect is harmonious. . . . They said 
that those bodies move most quickly which move at the greatest 
distance, that those bodies move most slowly which are at the 
least distance, and that the bodies at intermediate distances move 
at speeds corresponding to the sizes of their orbits.' 2 

We have no information as to the actual ratios which the Pytha- 
goreans assumed to exist between the respective distances of the 
earth, moon, sun, and planets from the centre of the universe. 
When Plutarch says that the distances of the ten heavenly bodies 
formed, according to Philolaus, a geometrical progression with 3 as 
the common ratio, 3 he can only be referring to some much later 
Pythagoreans. For if, on the basis of this progression, the distance 
of the counter-earth is represented by 3, that of the earth by 
9, and that of the moon by 27, it is obvious that the enormous 
parallaxes due to the revolution of the earth round the centre would 

1 Arist. De caelo ii. 9, 290 b 12-29 (Vors. i 2 , p. 277. 28-42). Yet when 
Aristotle is trying to prove his own contention that the stars do not move of 
themselves but are carried by spheres which revolve, he does not hesitate to use 
the argument that, if the planets moved freely through a mass of air or fire 
spread through the universe, 'as is universally alleged', they would, in conse- 
quence of their size, inevitably produce a sound so overpowering that it would not 
only be transmitted to us but would actually shiver things. He maintains, how- 
ever, that, if a body is carried by something else which moves continuously and 
does not cause actual concussion, it does not produce sound ; hence, in his view, 
the fact that we do not hear sounds from the motion of the planets implies that 
they have no motion of their own but are carried by something [De caelo ii. 9, 
291 a 16-28). 

2 Alexander on Metaph. A. 5, p. 542 a 5-10, 16-18 Brandis, pp. 39. 24-40. 1, 
40. 7-9 Hayduck. 3 Plutarch, De animae procreatione, c. 31, p. 1028 B. 



CH. χπ THE PYTHAGOREANS 107 

be quite inconsistent with ' saving the phenomena '.* Moreover, 
the order of the heavenly bodies given in this passage, counter- 
earth, earth, moon, Mercury, Venus, Sun, is not the order in 
which they were placed by Philolaus (and by Plato later) but the 
Chaldaean order, which does not seem to have been adopted by any 
Greek before the Stoic Diogenes of Babylon (second century B.C.). 
Of the 'harmony of the spheres' there are many divergent 
accounts, 2 and it would appear that the places and the number of 
the heavenly bodies supposed to take part in it varied at different 
periods. Burnet 3 suggests that we cannot attribute to Pytha- 
goras himself more than an identification of his newly-discovered 
musical intervals, the fourth, fifth, and octave, with the three rings 
which we find in Anaximander, that of the stars (nearest to the 
earth), that of the moon (next) and that of the sun (which is the 
furthest from the earth), and that this would be the most natural 
beginning for the later doctrine of the ' harmony of the spheres \ 
This is an attractive supposition, but it depends on the assumption 
that Pythagoras attributed to the planets and the fixed stars the 
same revolution from east to west ; whereas he certainly dis- 
tinguished the planets from the fixed stars, and he must have 
known that their movement was not the same as that of the fixed 
stars (this is clear from his identification of the Morning and 
Evening Stars), even if he did not assign to the planets the inde- 
pendent movement, in the opposite sense to the daily rotation, 
which Alcmaeon is said to have observed. The original form oT^ 
the theory of the ' harmony of the spheres ' no doubt had reference 
to the seven planets only (including in that term the sun and moon), 
the seven planets being supposed, by reason of their several motions, 
to give out notes corresponding to the notes of the Heptachord ; 4 

1 Schiaparelli, I precursori, pp. 6, 44. 

2 I must refer for full details to Boeckh, Studien iii, pp. 87 sqq. {Kleine 
Schrifien, iii, pp. 169 sq.), Carl v. Jan, Philol. 1893, pp. 13 sqq., and for a 
summary to Zeller, i 5 , pp. 431-4. 

3 Burnet, Early Greek Philosophy, p. 122. 

4 Cf. Hippol. Refut. i. 2. 2, (D. G. p. 555), ' Pythagoras maintained that the 
universe sings and is constructed in accordance with a harmony ; and he was 
the first to reduce the motion of the seven heavenly bodies to rhythm and song ' ; 
Censorinus, De die natali 13. 5, ' Pythagoras showed that the whole of our 
world constitutes a harmony. Accordingly, Dorylaus wrote that the world 
is an instrument of God ; others added that it is a heptachord, because there 
are seven planets which have the most motion.' 



io8 THE PYTHAGOREANS parti 

it could not have related to the ten heavenly bodies of the Pytha- 
gorean system, for this would have required ten notes, whereas the 
Pythagorean theory of tones only recognized the seven notes of 
the Heptachord. This may, as Zeller says, 1 be the reason why 
Philolaus himself, so far as we can judge from the fragments, said 
nothing about the harmony of the spheres. Aristotle, however, 
clearly implies that in the harmony of the Pythagoreans whom he 
knew the sphere of the fixed stars took part ; for he speaks of the 
intolerable noise which, on the assumption that the motion of the 
heavenly bodies produced sound, would be caused by ' the stars 
which are so many in number and so great: 2 Consequently eight 
notes are implied : and accordingly we find Plato (in Republic x) 
including in his harmony eight notes produced by the sphere 
of the fixed stars and by the seven planets respectively, and 
corresponding to the Octachord, the eight-stringed lyre which had 
been invented in the meantime. The old theory being that all the 
heavenly bodies revolved in the same direction from east to west, 
only the planets revolved more slowly, their speeds diminishing in 
the order of their distances from the sphere of the fixed stars, 
which rotates once in about 24 hours, it would follow that Saturn, 
being the nearest to the said sphere, would be supposed to move the 
most quickly; Jupiter, being next, would be the next quickest; 
Mars would come next, and so on ; while the moon, being the 
innermost, would be the slowest ; on this view, therefore, the note 
of Saturn would be the highest (νήτη), that of Jupiter next, and so 
on, that of the moon being the lowest (υπάτη) ; and the speeds 
determining this order are absolute speeds in space. Nicomachus, 3 
though he mentions that his predecessors assigned notes to the 
seven planets in this order, himself took the opposite view, 
placing the moon's tone as the highest and Saturn's as the lowest 
(incidentally he places the sun in the middle of the seven instead 
of next to the moon as the older system did). Nicomachus's order 
is explicable if we assume that the independent revolutions of the 
planets (in their orbits) was the criterion for the assignment of the 
notes; for the moon describes its orbit the quickest (in about 
a lunar month), the sun the next quickest (in a year), and so on, 
Saturn being the slowest in describing its orbit ; these speeds are 

1 Zeller,i 5 . p. 432, note 2. 2 Aristotle, De caelo\\. 9, 290b 18 (Vors. i 2 , p. 277. S3). 
3 Nicomachus, Hartn. 6. 33 sq. ; cf. Boethius, Inst. Mus. i. 27. 



CH.xn THE PYTHAGOREANS 109 

relative speeds, i.e. relative to the sphere of the fixed stars regarded 
as stationary. The absolute and relative angular speeds of the 
planets are of course connected in the following way : for any one 
planet its absolute speed is the speed of the sphere of the fixed stars 
minus the relative speed of the planet ; hence their order in respect 
of absolute speed is the reverse of their order in respect of relative 
speed and, so long as only the seven planets (including the sun 
and moon) come into the scale of notes, it is possible to assign 
notes to them in either order. But this is no longer the case when 
the sphere of the fixed stars is brought in as having a note of its 
own, making altogether eight notes corresponding to the Octachord. 
The speed of the fixed stars is of course an absolute speed, and it is 
faster than either the absolute or relative speed of any of the 
planets ; it must, therefore, give out the highest note (νήτη). Now, 
in assigning the rest of the notes, we cannot take the relative speeds 
of the planets for the purpose of comparison with the absolute 
speed of the sphere of the fixed stars ; we must compare like with 
like; and indeed, on the hypothesis that the body which moves 
more swiftly gives out a higher note than the body which moves 
more slowly, it is only the absolute speed of the heavenly bodies 
in space, and nothing else, which can properly be taken as deter- 
mining the order of their notes. Now Plato says 1 in the Myth of 
Er that eight different notes forming a harmony are given out by 
the Sirens seated on the eight whorls of the Spindle, which repre- 
sent the sphere of the fixed stars and the seven planets, and that, 
while all the seven inner whorls (representing the planets) are carried 
round bodily in the revolution of the outermost whorl (representing 
the sphere or circle of the fixed stars), each of the seven inner 
whorls has a slow independent movement of its own in a sense 
opposite to that of the movement of the whole, the second whorl 
starting from the outside (the first of the seven inner ones) which 
represents Saturn having the slowest movement, the third repre- 
senting Jupiter the next faster, the fourth representing Mars the 
next faster, the fifth, sixth, and seventh, which represent Mercury, 
Venus, and the Sun respectively and which go 'together' (i.e. have 
the same angular speed) the fastest but one, and the eighth repre- 
senting the moon the fastest of all. Plato, therefore, while speaking 

1 Plato, Republic x. 617 a-b. 



no THE PYTHAGOREANS parti 

of absolute angular speed in the case of the circle of the fixed stars, 
refers to the relative speed in the case of the seven planets. To 
get the order of his tones therefore we must turn the relative speeds 
of the planets into absolute speeds by subtracting them respectively 
from the speed of the circle of the fixed stars, and the order of 
their respective notes is then as follows : 

Circle of fixed stars . . . highest note (νήτη) 

Saturn 

Jupiter 

Mars 

Mercury") 

Venus L 

Sun ) 

Moon lowest note (ύπατη). 1 

1 Dr. Adam, in his edition of the Republic (vol. ii, p. 452), supposes that, after 
the circle of the fixed stars giving the highest note, the seven planets would 
come in the order of their relative velocities, thus — 

Circle of the fixed stars . . . highest note (νήτη) 

„ Moon 

„ Sun J 

„ Venus Υ . . . μίση 

„ Mercury j 

„ Mars 

„ Jupiter 

„ Saturn .... lowest note (νπάτη) 

For the reason given above, I do not think it possible that Plato, who was 
a mathematician, would have assigned the notes to the eight circles in this 
order, though it is likely enough that, when writing the passage, he had not 
in his mind any definite allocation of notes at all. A further difficulty in the 
way of Adam's order is the following. He observes that, if we understand 
' together ' (άμα άλλήλοις), used of the motion of the sun, Venus, and Mercury, 
in a strict sense, there will only be six notes, as the three bodies will have the 
same note. He gets over this difficulty quite properly by supposing that Plato 
really had in his mind the period taken by the three bodies in describing their 
orbits, in other words, their angular velocity, rather than their linear velocity. 
* In that case the octave will be complete, because, in order to complete their 
orbits in the same time, the sun, Venus, and Mercury will have to travel at 
different rates of speed.' True ; but, as the planet with the longer orbit must 
have a linear velocity greater than the planet with the shorter orbit, it follows 
that the linear velocity of Venus in the above scheme will be greater than that 
of the sun, and the linear velocity of Mercury greater than that of Venus. Thus 
the supposed linear velocities, instead of diminishing all the way from the circle 
of the fixed stars down to Saturn in the above table, will diminish from the 
circle of the fixed stars down to the sun, but will increase after that down to 
Mercury, before they diminish again with Mars and the rest; and this upsets 
the proper order of the notes altogether. On the other hand, with the arrange- 
ment according to absolute speeds, as in the text above, the linear velocities of 
Mercury, Venus, and the sun come in the correct diminishing order. 



CH.xn THE PYTHAGOREANS jii 

This order agrees with Cicero's arrangement, in which the highest 
circle, that of the fixed stars, has the highest note and the moon 
the lowest. 1 

Although Alexander clearly says that, in the Pythagorean theory 
of the harmony of the spheres, the different notes correspond to 
the ratios of the distances of the heavenly bodies, we have little 
or no authentic information as to how the early Pythagoreans 
translated the theory into an actual estimate of the relative 
distances. 2 It is true that some later writers such as Censorinus 
and Pliny give some definite ratios of distances and, as usual, refer 
them back to Pythagoras himself; but their statements contain 
such an admixture of elements foreign to the early Pythagorean 
theory that no certain conclusion can be drawn. 

Plato implies, in his Myth of Er, that the breadths of the whorls 
of the spindle represent the distances separating successive planets, 
but he does not do more than state the order of magnitude in 
which the successive distances come ; he makes no attempt to give 
absolute ratios between them. 

Tannery 3 ingeniously conjectures that Eudoxus's view of the 
ratio of the distances of the sun and moon from the earth, which 
he put at 9:1, may have been suggested or confirmed by the 
theory of the harmony. The original discovery of the octave, 
the fourth and the fifth, stated in one of its forms, 4 showed that 
they represented ratios of lengths of string assumed to be under 
equal tension as follows, namely 1:3, 3:4, and 2 : 3 respectively. 
Bringing these ratios to their least common denominator, we see 
that strings at equal tension and of lengths 6, 8, 9, 12 respectively 
give the three intervals. The interval between the first and second 
strings being a fourth, and that between the first and third a fifth, 
the interval between the second and third is a tone, which may 
therefore be regarded as represented by the difference between 

1 Cicero, Somn. Scip. c. 5. 

2 Alexander's own figures (Alex, on Metaph. 986 a 2, p. 542 a 12-15 Brandis, 
p. 40. 3-6 Hayduck) seem to be illustrations only : ' The distance of the sun 
from the earth being, say (φερ? etVeii/), double the distance of the moon, that of 
Aphrodite triple, and that of Hermes quadruple, they considered that there was 
some arithmetical ratio in the case of each of the other planets as well.' The 
ratios of 1, 2, 3, 4 for the distances of the moon, the sun, Venus, and Mercury 
are the same as those indicated by Plato in the Timaeus 36 D. 

3 Tannery, Recherches sur Vhistoire de V astronomie ancienne, pp. 293, 328. 

4 Cf. Theon of Smyrna, pp. 59. 21-60. 6, ed. Hiller ; Boethius, hist. Mus. i. 10. 



ii2 THE PYTHAGOREANS parti 

9 and 8, or I. Now the Didascalia caelestis of Leptines, known as 
Ars Endoxi, which was written in Egypt between 193 and 165 β.α, 
contains a number of things derived from Eudoxus, and the ratio 
of the distance of the sun from the earth to the distance of the moon 
from the earth is there said to correspond to the relation of the fifth 
to the tone. 1 If we take the respective notes as represented by the 
above numbers, the ratio of the fifth to the tone is 9 : (9 — 8), or 9 : 1. 

It would appear from passages in Theon of Smyrna 2 and 
Achilles, 3 doubtless taken in substance from Adrastus or Thrasyllus, 
that the harmony was next spoken of in poems by Aratus and 
Eratosthenes (third century B.C.) ; but there is no indication that 
they did more than point out the correspondence between the 
planets, in their order from the moon to Saturn or to the sphere 
of the fixed stars, and the notes of the heptachord or octachord 
from the ύπατη, the lowest, to the νήτη, the highest (Eratosthenes 
certainly took the octachord for this purpose). 4 

Achilles tells us that, after Aratus and Eratosthenes, and before 
Adrastus and Thrasyllus, Hypsicles the mathematician (the author 
of the so-called Book XIV of Euclid) treated of the question of 
the harmony of the spheres ; and he proceeds to give, as generally 
accepted by musicians, a remarkable musical scale in which an 
octave is divided into eight intervals and nine notes (including the 
two extreme notes of the octave), the nine notes corresponding 
to the sphere of the fixed stars, Saturn, Jupiter, Mars, Mercury, 
Venus, Sun, Moon, and Earth respectively, in that order. This 
scale is the same as that described in verses quoted by Theon 
of Smyrna from one Alexander (who was not Alexander of 
Aetolia, as Theon wrongly calls him, but Alexander of Ephesus, 
a contemporary of Cicero, or possibly, as Chalcidius calls him, 
Alexander of Miletus, Alexander Polyhistor). The only difference 

1 The text, indeed, of Leptines has to be filled out in order to get this, and it 
is the sizes of the sun and moon, not their distances respectively from the earth, 
that are mentioned (though the effect is the same on the assumption that their 
apparent angular diameters are equal). The sentence as corrected by Tannery 
is 'Thus the sun is greater than the moon, and the moon greater than (the 
part of) the earth (which sees the eclipse); the ratio is that of the fifth to (the 
difference between the fifth and) the fourth.' 

2 Theon of Smyrna, pp. 105. 13-106. 2 ; pp. 142. 7 sqq. 

8 Petav. Uranolog. p. 136; see Tannery, Recherches sur V histoire de Γ astro- 
nomie ancienne, p. 330. 

4 Theon of Smyrna, loc. cit. 



ch.xii THE PYTHAGOREANS 113 

is that Alexander has the later order for the planets, his order 
being: sphere of fixed stars, Saturn, Jupiter, Mars, Sun, Venus, 
Mercury, Moon, Earth. 1 Tannery infers that this peculiar division 
of the octave, with the order of the planets as given by Achilles, 
is due to Hypsicles. 2 

Theon of Smyrna criticizes this peculiar scale of nine notes as 
described by Alexander. First, he observes that in the last of 
the verses Alexander says the heptachord is the image of the 
world, whereas he has made an octave, consisting of six tones, out 
of nine strings ; his notes therefore do not correspond to the 
diatonic scale. Again, the lowest note is given to the earth, 
whereas, being at rest, it gives out no sound. The sun, too, is 
given the 'middle' note (μύση), whereas the interval from the 
lowest (υπάτη) to the * middle ' is not a fifth but a fourth ; and 
so on. 

The scale, however, of nine notes with the sun in the middle, 
as Alexander has it, is apparently the common foundation of three 
scales of eight intervals given by Censorinus, 3 Pliny, 4 and Martianus 
Capella 5 respectively, who apparently got them from a work of 
the encyclopaedic writer Varro (116-27 B.C.). The three scales 
given by these three authors differ slightly in that Censorinus's 
eight intervals add up to 6 tones (the proper amount), Pliny's to 
7 tones, and Martianus Capella's to 6\ tones ; the differences may, 
Tannery thinks, be due to errors in the MSS. of Varro, whence 
the one scale which is the foundation of all three was taken. We 
need only set down Censorinus's version, which is : 

From Earth to Moon 1 tone\ 



-3§ tones (a fifth) 



„ Moon to Mercury \ n 

„ Mercury to Venus \ , 

,, Venus to Sun ij „ 

„ Sun to Mars 1 „ \ 

„ Mars to Jupiter \ „ , Λ . - .,. 

" T . J 1 λ \ί\ tones (a fourth) 

„ Jupiter to Saturn i » 

„ Saturn to fixed stars f „ 

6 tones 

1 Theon of Smyrna, pp. 140. 5-141. 4. 2 Tannery, loc. cit. 

3 Censorinus, De die natali 13. 3-5. 4 Pliny, Ν. H. ii, c. 22, § 84. 

5 Mart. Capella, De nuptiis philologiae et Mercurii, ii. 169-98. 

1410 I 



u 4 THE PYTHAGOREANS parti 

The difference between this and Pliny's scale is that Pliny takes 
the distance from Saturn to the sphere of the fixed stars to be 
ij tones instead of half a tone, so that with him the distance 
between the sun and the fixed stars is 3I tones, or a fifth instead 
of a fourth. Both Censorinus and Pliny make the interval from the 
earth to the sun to be a fifth, and from the earth to the moon one tone, 
wherein they agree with the view attributed by Tannery to Eudoxus. 

Both Pliny and Censorinus add a further detail which apparently 
must have come from some source other than the poem of 
Alexander ; this is that Pythagoras made the actual distance 
between the moon and the earth, which he called one tone, to 
be 1 26,000 stades. This would of course enable the other distances 
between the heavenly bodies to be calculated on the basis of the 
scale ; e. g. the distance from the earth to the sun would be 3^ 
times 126,000 stades, and so on. But this evaluation of the 
distance from the earth to the moon, 126,000 stades, is exactly 
half of 252,000 stades, which is the estimate of the circumference 
of the earth made by Eratosthenes and Hipparchus. This exact 
coincidence is enough to make it plain that the 126,000 stades 
does not go back to Pythagoras, and can hardly have been 
suggested before the second century B. C. 

Pliny, however, in a passage immediately preceding that in 
which he describes his scale, says that Pythagoras made the 
distance from the earth to the moon 126,000 stades, the distance 
from the moon to the sun twice that distance, and the distance from 
the sun to the sphere of the fixed stars thrice the same distance. 1 
Pliny is here evidently quoting from a quite different authority ; 
as he says that Sulpicius Gallus was of the same opinion, he would 
appear to be citing some book by Sulpicius Gallus, who may have 
got it from some tradition which cannot now be traced. 

It is no doubt possible that, if Pythagoras did not estimate the 
distance of the moon from the earth in stades, he may have 
expressed it in terms of the circumference of the earth. But, seeing 
that Anaximander had already estimated the radius of the orbit of 
the moon at 18 times the radius of the earth, how could Pythagoras 
have put the distance of the moon so low as half the circumference 
of the earth, or about 3 times the earth's radius? Tannery 

1 Pliny, N.H. ii, c. 21, §83. 



ch.xii THE PYTHAGOREANS 115 

conjectures that in the number of stades (126,000) given by Varro 
there is a mistake, milia having been written instead of myriads 
(μυριάδας) ; in that case the source from which Varro drew might 
have given the distance of the moon as 10 times the half- 
circumference of the earth. Hultsch, 1 however, thinks it incredible 
that milia could have been written in error for μυριάδας ; and even 
if it had been, and the moon's distance were thus made up to about 
30 times the earth's radius, the absurdity would still be left that 
the sun's distance is only 3^ times as great. 

It is true, as Martin observes, 2 that the sounding by the planets 
of all the notes of an octave at once would produce no ' harmony ' 
in our sense of the word ; but the Pythagoreans would not have 
been deterred by this consideration from putting forward their 
fanciful view. 3 We have, it is true, allusions to other arrangements 
of the notes which would make them cover more than an octave, 
but these must have been later than Plato's time. Thus Plutarch 
speaks of one view which made the seven planets correspond to 
the seven invariable strings of the fifteen-stringed lyre, and of 
another which made their distances correspond to the five tetra- 
chords of the complete system. 4 Anatolius 5 has a peculiar 
distribution of tones between the heavenly bodies which gives 
altogether two octaves and a tone. Macrobius 6 bases his view 
on the successive numbers 1, 2, 3, 4, 8, 9, 27 applied to the planets 
in the Timaeus and supposed to represent their relative distances 
from the earth ; Macrobius makes the first four (from 1 to 4) cover 
two octaves, and he seems to make the seven notes cover, in all, 
four octaves, a fifth, and one tone. 7 

The Sun. 

'The Pythagoreans declared the sun to be spherical.' 8 
1 Philolaus the Pythagorean holds that the sun is transparent like 
glass, and that it receives the reflection of the fire in the universe 

1 Hultsch, Poseidonios iiber die Grosse und Entfernung der Sonne , Berlin, 
1897, p. 11, npte 1. 

2 Martin, Etudes sur le Time'e, ii, p. yj. 3 Zeller, i 5 , p. 432, note. 

4 Plutarch, De animae firocr. c. 32, p. 1029 a, B. The five distances are 
(1) Moon to Sun with its concomitants Mercury and Venus, (2) Sun, &c. to Mars, 
(3) Mars to Jupiter, (4) Jupiter to Saturn, (5) Saturn to sphere of fixed stars. 

5 Anatolius in Iambi. Theol. Ar. p. 56 ; cf. Zeller, loc. cit. 

6 Macrobius, In Somn. Scifi. ii, cc. 1, 2. 

7 Zeller, ii 4 , pp. ηηη sqq. 8 Aet. ii. 22. 5 (D. G. p. 352). 

I 2 



n6 THE PYTHAGOREANS parti 

and transmits to us both light and warmth, so that there are in 
some sort two suns, the fiery (substance) in the heaven and the fiery 
(emanation) from it which is mirrored, as it were, not to speak of 
a third also, namely the beams which are scattered in our direction 
from the mirror by way of reflection (or refraction); for we give 
this third also the name of sun, which is thus, as it were, an image 
of an image.' 1 

1 Philolaus says that the sun receives its fiery and radiant nature 
from above, from the aethereal fire, and transmits the beams to us 
through certain pores, so that according to him the sun is triple, 
one sun being the aethereal fire, the second that which is transmitted 
from it to the glassy thing under it which is called sun, and the 
third that which is transmitted from the sun in this sense to us.' 2 

Thus, according to Philolaus, the sun was not a body with light 
of its own, but it was of a substance comparable to glass, and it 
concentrated rays of fire from elsewhere, and transmitted them to 
us. This idea was no doubt suggested in order to give a uniform 
nature to all the moving heavenly bodies. But there are difficulties 
in the descriptions above given of the sources of the beams of 
fire. The natural supposition would be that they would come 
from the central fire ; in that case the sun would act like a mirror 
simply; and the phenomena would be accounted for because the 
beams of the fire would always reach the sun except when ob- 
structed by the moon, earth, or counter-earth, and, as the earth and 
counter-earth move in a different plane from the sun and moon, 
eclipses would occur at the proper times. But the first of the above 
passages says that the beams come from the fire in the universe, 
and that one of the suns is the fiery substance in the heaven, while 
the second passage says that the beams come from above, from the 
fire of the aether. Burnet takes ' heaven ' in the narrow sense of the 
1 portion of the universe below the moon and round about the earth ' 
which, according to the Doxographi, was called ' heaven ', 3 and he 
thinks that 'the fire in the heaven' is therefore exclusively the 
central fire. 4 But this leaves out of account the alternative term 
4 the fire in the universe ' and also Achilles' ' fire from above ' ; and, 

1 Aet. ii. 20. 12 {D. G. p. 349, 350; Vors. i 2 , p. 237. 36). 

2 Achilles, Isagoge in phaenomena (Petav. Uranolog, p. 138; D. C7. 
PP• 349, 35o). 

s See above, p. 97 (Aet. ii. 7. 7 ; D. G. p. 337 ; Vors. i 2 , p. 237. 22). 
4 Burnet, Early Greek Philosophy, p. 348. 









CH.xn THE PYTHAGOREANS 117 

as the central fire seems in other passages always to be called ' the 
fire in the middle ', Burnet's interpretation seems scarcely possible. 
Boeckh originally took the same view that the beams could only 
be those from the central fire, holding to the strict interpretation of 
universe as being below the outer Olympus j 1 but he afterwards 
admitted, 2 with Martin, that the beams might come from the outer 
fire, the fire of Olympus, as well. Accordingly the beams coming 
from outside would be refracted by the sun, which would act as 
a sort of lens. 3 Tannery 4 takes a similar view, from which he 
develops another interesting hypothesis. We are to suppose two 
cones opposite to one another and each truncated at the sun, where 
they meet in a common section ; these two cones form a luminous 
column (that of the Myth of Er) by which a stream of light flows 
from the fire of Olympus (supposed to be the Milky Way) in the 
direction of the earth. But there remains a difficulty as regards the 
central fire. What is the relation between the central fire and 
the fire of the sun, and why does not the central fire always light 
up the moon sufficiently for us to see it full ? The beams of the 
central fire must, Tannery conceives, be relatively feeble in com- 
parison with those from the Milky Way, and though they may 
suitably light up and warm the side of the counter-earth turned 
towards the central fire, they have no appreciable power at the 
distance of the moon, still less at the distance of the sun. The 
outer cone and the inner cone meeting at the sun are supposed 
by Tannery to have a small angular aperture. The base of the 
outer cone is therefore presumably a part of the Milky Way, 5 
which part is accordingly the first sun of the texts, and Tannery 
suggests that we have in this portion of the Milky Way the tenth of 
the heavenly bodies which revolve round the central fire, leaving the 
sphere of the fixed stars motionless, as the complete system of 
Philolaus requires it to be. This suggestion is brilliant but scarcely, 
I think, consistent with what we are told of the tenth body ; for 

1 Boeckh, Philolaus, pp. 123-30. 

2 Boeckh, Das kos?nische System des Platon, p. 94. 

3 Martin, Uhypothese asironomique de Philolaus, pp. 9, 10. 

4 Tannery, Pour Vhistoire de la science hellkne, pp. 237, 238. 

5 Cf. Aet. i. 14. 2 (D. G. p. 312), where it is stated that only the fire in the 
very uppermost place is conical. The passage occurs in a section dealing mainly 
with the shapes of the elements, but it may perhaps have strayed into the wrong 
context. 



n8 THE PYTHAGOREANS parti 

on this assumption it would presumably be, from time to time, 
a different portion of the Milky Way varying as the sun revolves. 

With Tannery's idea of the connexion between the sun and the 
Milky Way, the following passages should be compared : 

' Of the so-called Pythagoreans some say that this [the Milky 
Way] is the path of one of the stars which fell out of their places 
in the destruction said to have taken place in Phaethon's time ; 
others say that the sun formerly revolved in this circle, and accord- 
ingly this region was, so to say, burnt up, or suffered some such 
change, through the revolution of the sun.' 1 

' Of the Pythagoreans some explain the Milky Way as due to the 
burning-up of a star which fell out of its proper place and set on fire 
the region through which it circulated during the conflagration 
caused by Phaethon ; others say that the sun's course originally lay 
along the Milky Way. Some, again, say that it is the mirrored 
image of the sun as it reflects its rays at the heaven, the process 
being the same as with the rainbow on the clouds.' 2 

The Moon. 

A fanciful view of the moon is quoted by the Doxographi as 
held by some of the Pythagoreans, including Philolaus. 

1 Some of the Pythagoreans, among whom is Philolaus, say that 
the moon has an earthy appearance because, like our earth, it is 
inhabited throughout by animals and plants, only larger and more 
beautiful (than ours) : for the animals on it are fifteen times stronger 
than those on the earth . . . and the day in the moon is correspond- 
ingly longer.' 3 

No doubt the fact that the animals on the moon are superior to 
those on the earth f in force (rfj δυνάμει) ' to the extent of fifteen 
times is an inference from the fact that the day is fifteen times 
longer than ours. Boeckh points out, as regards the day, that the 
length of it is clearly meant to be half the time occupied by one 
revolution of the moon (in 29I days) round the central fire. Assum- 
ing that, as with the earth, the same hemisphere is always turned 
outwards (which involves one rotation of the moon round its axis in 

1 Aristotle, Meteorologica i. 8, 345 a 13-18 (Vors. i 2 , p. 230. 37-41). In the 
last words of this passage Diels (loc. cit.) reads φθοράς, 'destruction' or 
' wasting ', instead of φοράς, ' revolution '. 
. 2 Aet. iii. 1. 2 (D. G. p. 364 ; Vors. i 2 , p. 278. 42). 
5 Aet. ii. 30. 1 (D. G. p. 361 ; Vors. i 2 , p. 237. 42). 



CH.xn THE PYTHAGOREANS 119 

the same time as it takes the moon to revolve round the central fire), 
an inhabitant of that hemisphere would see the sun, that is, it would 
be day for him, for roughly half the period of the moon's revolu- 
tion ; during the same half of the period an inhabitant of the 
hemisphere turned towards the earth would not see the sun, and it 
would be night for him ; and vice versa. Therefore the ' day ' for 
an inhabitant of the moon, which receives its light from the sun, 
would be equal to fifteen of our days and nights added together. 
According to the actual wording of the text it should be fifteen 
times our day only ; this would require that the moon should 
revolve on its axis twice (instead of the once which is automatic, 
as it were) during a lunation. Martin x develops this supposition, 
but it seems clear that the ' day ' of the inhabitants of the moon 
was meant to be equal to fifteen of our days and nights together, 
and that the form of the statement in the text is due to 
inadvertence. 

According to 'other Pythagoreans' what we see on the moon 
is a reflection of the sea which is beyond the torrid circle or zone in 

our earth. 2 

Eclipses. 

We have seen that the counter-earth was probably invented in 
order to explain the frequency of eclipses of the moon, and that 
there were some who thought there might be more bodies of the 
kind which by their interposition caused eclipses of the moon. The 
latter bodies would of course, like the counter-earth, be invisible to 
the inhabitants of our hemisphere, from which it follows that they 
would also, like the counter-earth, revolve along with the earth round 
the central fire and always have the same right ascension with 
the earth. 

Eclipses of the moon are then caused by the interposition either 
of the earth or of the counter-earth (or other similar body) between 
the sun and the moon. 3 

Eclipses of the sun on the other hand are, and can only be, caused 
by the moon ' getting under the sun ', 4 i.e. by the interposition of 
the moon between the sun and the earth. 

1 Martin, Hypothese astronomique de Ph'/olaus, p. 22. 

2 Aet. ii. 30. ι (Ζλ G. p. 361 b 10-13). 

3 Aet. ii. 29. 4 (D. G. p. 360 ; Vors. i 2 , p. 277. 46). 

4 σελήνης αυτόν υπερχομένης, Aet. ii. 24. 6 {D. G. p. 354.)• 



ΐ2θ THE PYTHAGOREANS 

The Phases of the Moon. 

In the same passage (under the heading ' On the Eclipse of the 
Moon ') in which Aetius says that ' some of the Pythagoreans ' give 
the explanation of lunar eclipses just referred to, a curious view 
is mentioned as having been held by 'some of the later (Pytha- 
goreans) '. The words must apparently (notwithstanding their 
context) refer to the phases^ and not to eclipses, of the moon ; the 
change is said to come about ' by way of spreading of flame, which 
is kindled by degrees and in a regular manner until it produces the 
perfect full moon, after which again the flame is curtailed by cor- 
responding degrees until the conjunction, when it is completely 
extinguished \ It would seem that these ' later ' Pythagoreans had 
forgotten the fact that the moon gets its light from the sun, or at 
least had no clear understanding of the way in which the variations 
in the positions of the sun and moon relatively to the earth produce 
the variations in the shape of the portion of the illuminated half 
which is visible to us from time to time. 



XIII 

THE ATOMISTS, LEUCIPPUS AND DEMOCRITUS 

LEUCIPPUS of Elea or Miletus (it is uncertain which 1 ) was 
a contemporary of Anaxagoras and Empedocles ; and Democritus 
of Abdera was also a contemporary of Anaxagoras, though younger, 
for he was, according to his own account, 2 ' young when Anaxagoras 
was old ', from which it is inferred that he was born about 460 B.C. 
The place of the two Atomist philosophers in the history of astronomy 
is not a large one, for they made scarcely any advance upon their 
predecessors ; most of the views of Democritus are a restatement of 
those of Anaxagoras, even down to the crudest parts of his doctrine. 
As Burnet 3 says, the primitive character of the astronomy taught 
by Democritus as compared with that of Plato is the best evidence 
of the value of the Pythagorean researches. The weakness of 
Democritus's astronomy is the more remarkable because we have 
conclusive evidence that he was a really able mathematician. 
Archimedes 4 says that Democritus was the first to state that the 
volumes of a cone and a pyramid are one-third of the volumes 
of the cylinder and prism respectively which have the same base 
and height, though he was not able to prove these facts in the 
rigorous manner which alone came up to Archimedes' standard 
of what a scientific proof should be (the discovery of the proofs 
of the propositions by the powerful ' method of exhaustion ' was 

1 Simplicius in Phys. p. 28. 4 (from Theophrastus) ; see D. G. p. 483 ; Vors. i 2 , 
p. 344. 46. 

2 Diog. L. ix. 41 (Vors. i 2 , p. 387. 12). 

3 Burnet, Early Greek Philosophy, p. 345. 

4 Heiberg, * Eine neue Archimedes-Handschrift ' in Hermes, xlii, 1907, 
pp. 245, 246; cf. the translation and commentary by Heiberg and Zeuthen 
in Bibliotheca Mathematica, vii s , 1906-7, p. 323 ; The Thirteen Books of 
Euclid's Elements, 1908, vol. iii, pp. 366, 368. 



122 THE ATOMISTS parti 

reserved for Eudoxus). There is evidence, too, that Democritus 
investigated (i) the relation in size between two sections of a cone 
parallel to the base and very close to each other, and (2) the nature 
of the contact of a circle or sphere with a tangent. These facts 
taken together suggest that he was on the track of infinitesimals 
and of the Integral Calculus. 

The Great Diakosmos, attributed by Theophrastus to Leucippus, 
is also given in the lists of Democritus's works; 1 indeed no one 
later than Theophrastus seems to have been able to distinguish 
between the work of Leucippus and Democritus, all the writings 
of the school of Abdera being apparently regarded by later authors 
as due to Democritus. However, the information which we possess 
about the cosmology of the two philosophers goes back to Theo- 
phrastus, so that we are not without some guidance as to details 
in which they differed. Diogenes Laertius, 2 in a passage drawn 
from an epitome of Theophrastus, attributes the following views 
to Leucippus. The worlds, unlimited in number, arise through 
' bodies', i.e. atoms, falling into the void and meeting one another. 
By abscission from the infinite many ' bodies ' of all sorts of shapes 
are borne into a great void, and their coming together sets up 
a vortex. By the usual process, in the case of our world, the earth 
collects at the centre. The earth is like a tambourine in shape and 
rides or floats by virtue of its being whirled round in the centre. 

The sun revolves in a circle, as does the moon ; the circle of the 
sun is the outermost, that of the moon the nearest to the earth, and 
the circles of the stars are between. All the stars are set on fire 
because of the swiftness of their motion ; the sun is also ignited 
by the stars ; the moon has only a little fire in its composition. 

The ' inclination of the earth ', 3 i. e. the angle between the zenith 

1 Vors. i 2 , p. 357. 21, p. 387. 4; cf. Achilles, Isagoge i. 13 (Vors. i 2 , 
p. 349. 29). 

2 Diog. L. ix. 30-33 (Vors. i 2 , pp. 342. 35 -343. 27). 

The words ' inclination of the earth ' are missing in the text of Diogenes. 
Diels ( Vors. i 2 , p. 343. 22) supplies words thus: (την δε λόξωσιν τον ζωδιακού 
•γενέσθαι) τω κεκλίσθαι την γην προς μεσημβρίαν, ' (the obliquity of the zodiac 
circle is due) to the tilt of the earth towards the south.' But this can hardly 
be right ; the reference must be to the same ' inclination of the earth (έγκλισι? 
yrjs), i.e. the angle between the zenith and the pole or between the earth's (flat) 
surface and the plane of the apparent circular revolution of a star, which is 
spoken of in Aet. iii. 12. 1-2 (D. G. p. 377 ; Vors. i 2 , pp. 348. 15, 367. 47). The 
words which have fallen out may perhaps have been ' the obliquity of the circles 



ch.xtii LEUCIPPUS AND DEMOCRITUS 123 

and the visible (north) pole, or the angle between the (flat) surface 
of the earth and the plane of the apparent circular movement of 
a star in the daily rotation, is due to the tilt of the earth towards 
the south, the explanation of this tilt being on lines which recall 
Empedocles rather than Anaxagoras ; * the northern parts have 
perpetual snow and are cold and frozen. 

The sun rarely suffers eclipse, while the moon is continually 
darkened, because their circles are unequal. 

We have here reminiscences of Anaximander in the description 
of the shape of the earth and partly also in the statement about 
the relative distances of the sun, moon, and stars from the earth, 
while the idea of the earth riding on the air recalls Anaximenes, 
with a difference. There are traces of Anaxagoras's views in the 
vortex causing the earth to take the central position, and in the 
kindling of the stars due to their rapid motion ; but there is the 
difference that the atoms take the place of the mixture in which 
1 all things are together ', and no force such as Anaxagoras's Nous 
is considered to be required in order to start the motion of the 
vortex, the atoms being held to have been in motion always. 

Democritus's views are much more uniformly those of Anaxagoras. 
Thus with him the stars are stones, 2 the sun is a red-hot mass or 
a stone on fire ; 3 the sun is of considerable size. 4 The moon has 
in it plains, mountains (or, according to one passage, lofty elevations 
casting shadows 5 ), and ravines, 6 or valleys. 5 Democritus said that 
the moon is \ plumb opposite ' to the sun at the conjunctions, and 

of the stars ', or they may have referred to differences of climate in different 
parts of the earth. 

1 Leucippus's explanation of the tilt (Aet., loc. cit.) is that ' the earth turned 
sideways towards the southern regions because of the rarefaction (αραιότητα) in 
those parts, due to the fact that the northern regions became frozen through 
excessive cold while the southern parts were set on fire'. 

Democritus's explanation is slightly different : ' The earth as it grew became 
inclined southwards because the southern portion of the enveloping (substance) 
is weaker (i. e. presumably weaker in resisting power) ; for the northern regions 
are intemperate (Άκρατα), i.e. frigid, the southern temperate (κέκραταή ; hence 
it is in the south that the earth sags (βφάρηται), namely, where fruits and all 
growth are in excess.' 

Aet. ii. 13. 4 (D. G. p. 341 ; Vors. i 2 , p. 366. 31). 

3 Aet. ii. 20. 7 (D. G. p. 349 ; Vors. i 2 , p. 366. 35). 

4 Cicero, De fin. i. 6. 20 (Vors. i 2 , p. 366. 36). 

5 Aet. ii. 30. 3 (D. G. p. 361 ; Vors. i 2 , p. 367. 13). 

6 Aet. ii. 25. 9 (D. G. p. 356; Vors. i 2 , p. 308. 11). 



2 



124 



THE ATOMISTS parti 



it is evident that he fully accepted the doctrine that the moon 
receives its light from the sun. 1 

As regards the earth, Democritus differed from Anaxagoras in 
that, while Anaxagoras said it was flat, Democritus regarded it as 
'disc-like but hollowed out in the middle' 2 (i.e. depressed in the 
middle and raised at the edges) ; but this latter view was also held 
by Archelaus, a disciple of Anaxagoras, and may therefore have 
been that of Anaxagoras himself; the proof of the hollowness, 
Archelaus thought, was furnished by the fact that the sun does not 
rise and set everywhere on the earth's surface at the same time, 
as it would have been bound to do if the surface had been 
level. 3 How, asks Tannery, 4 did Anaxagoras or Archelaus come 
to draw from the observed facts with regard to the rising and setting 
of the sun a conclusion the very opposite of the truth ? 

Again, while Anaxagoras, like Anaximenes, supposed the flat 
earth to ride on the air, being supported by it, 5 Democritus is 
associated with Parmenides' view that the earth remains where 
it is because it is in equilibrium and there is no reason why it 
should move one way rather than another. 6 

We are told that the ancients represented the inhabited earth 
as circular, and regarded Greece as lying in the middle of it and 
Delphi as being in the centre of Greece, but that Democritus was 
the first to recognize that the earth is elongated, its length being 
i| times its breadth. 7 Democritus is also, along with Eudoxus, 
credited with having compiled a geographical and nautical survey 
of the earth as, after Anaximander, Hecataeus of Miletus and 
Damastes of Sigeum had done. 8 

Democritus agreed with Anaxagoras's remarkable view of the 
Milky Way as consisting of the stars which the sun ' does not see ' 

1 Plutarch, De facie in orbe lunae 16, p. 929 c (Vors. i 2 , p. 367. 9-1 1). 
Plutarch is arguing that the moon is made of an opaque substance, like earth. 
Were it otherwise, he says, the moon would not be invisible at the conjunctions 
when 'plumb opposite' the sun; if, e.g., the moon were made of a transparent 
material like glass or crystal, then, at the conjunctions, it should not only be 
visible itself, but it should allow the sun's light to shine through it, whereas it is 
in fact invisible at those times and often actually hides the sun from our sight. 

2 Aet. iii. 10. 5 (D. G. p. 377 ; Vors. i 2 , p. 367. 41). 

8 Hippolytus, Refut. i. 9. 4 {D. G. pp. 563-4; Vors. i 2 , p. 324. 16). 

4 Tannery, Pour fhistoire de la science hellene, p. 279. 

5 Hippol. Refut. i. 8. 3 (D. G. p. 562. 5-7 ; Vors. i 2 , p. 301. 32). 

6 Aet. iii. 15. 7 {D. G. p. 380; Vors. i 2 , p. in. 40). 

7 Agathemerus, i. 1. 2 (Vors. i 2 , p. 393. 10). 8 Ibid. 



ch. χιπ LEUCIPPUS AND DEMOCRITUS 125 

when it is passing under the earth during the night ; * but, at the 
same time, he seems to have been the first to appreciate its true 
character as a multitude of small stars so close together that the 
narrow spaces between them seem even to be covered by the 
diffusion of their light in all directions, so that it has the appearance, 
almost, of a continuous body of light. 2 

With Anaxagoras he thought that comets were ' a conjunction 
of planets when they come near and appear to touch one another ', 3 
or a f coalescence of two or more stars so that their rays unite \ 4 

In his remark, too, about the infinite number of worlds he seems 
to have done little more than expand what Anaxagoras had said 
about the men in other worlds than ours who have inhabited 
cities and cultivated fields, a sun and moon of their own, and 
so on. 5 It is worth while to quote Democritus's actual words in 
full, in order to see how slight is the foundation for the rhapsodical 
estimate which Gomperz gives of his significance as a forerunner 
of Copernicus. Hippolytus relates of Democritus that 

1 He said that there are worlds infinite in number and differing 
in size. In some there is neither sun nor moon, in others the sun 
and moon are greater than with us, in others there are more than 
one sun and moon. The distances between the worlds are unequal, 
in some directions there are more of them, in some fewer, some are 
growing, others are at their prime, and others again declining, in one 
direction they are coming into being, in another they are waning. 
Their destruction comes about through collision with one another. 
Some worlds are destitute of animal and plant life and of all 
moisture. ... A world is at its prime so long as it is no longer 
capable of taking in anything from without.' 6 

Let us now hear Gomperz. 7 ' Democritus's doctrine was far from 
admitting the plausible division of the universe into essentially 
different regions. It recognized no contrast between the sublunary 
world of change and the changeless steadiness of the divine stars, 
important and fatal though that difference became in the Aristotelian 

1 Aristotle, Meteorologica i. 8, 345 a 25 ( Vors. i 2 , p. 308. 26). 

2 Macrobius, In Somn. Scip. i. 15. 6 ; Aet. iii. 1. 6 (D. G. p. 365 ; Vors. i 2 , 

p. 367. 2.)• 

s Aristotle, Meteorologica i. 6, 342 b 27 ( Vors. i 2 , p. 308. 34). 

4 Aet. iii. 2. 2 {D. G. p. 366 ; Vors. i 2 , p. 308. 37). 

5 Anaxagoras, Fr. 4 ( Vors. i 2 , p. 315. 8-16). 

8 Hippolytus, Refut. i. 13. 2-4 [D. G. p. 565 ; Vors. i 2 , p. 360. 10-19). 
7 Gomperz, Griechische Denker, i 3 , pp. 295, 296. 



126 THE ATOMISTS parti 

system. At this point Democritus was once more fully in agree- 
ment not merely with the opinions of great men like Galilei, who 
released modern science from the fetters of Aristotelianism, but even 
with the actual results of the investigation of the last three centuries. 
It is almost miraculous to observe how the mere dropping of the 
scales from his eyes gave Democritus a glimpse of the revelations 
which we owe to the telescope and to spectrum analysis. In 
listening to Democritus, with his accounts of an infinitely large 
number of worlds, different in size, some of them attended by 
a quantity of moons [why not suns too, as in the fragment?], others 
without sun or moon, some of them waxing and others waning after 
a collision, others again devoid of every trace of fluid, we seem to 
hear the voice of a modern astronomer who has seen the moons of 
Jupiter, has recognized the lack of moisture in the neighbourhood 
of the moon, and has observed the nebulae and obscured stars which 
the wonderful instruments that have now been invented have made 
visible to his eyes. Yet this consentaneity rested on scarcely any- 
thing else than the absence of a powerful prejudice concealing the 
real state of things, and on a bold, but not an over-bold, assump- 
tion that in the infinitude of time and space the most diverse 
possibilities have been realized and fulfilled. So far as the endless 
multiformity of the atoms is concerned, that assumption has not 
won the favour of modern science, but it has been completely 
vindicated in respect to cosmic processes and transformations. It 
may legitimately be said that the Democritean theory of the 
universe deposed in principle the geocentric point of view. Nor 
would it be unfair to suppose that Democritus smoothed the way 
for its actual deposition at the hands of Aristarchus of Samos/ . . . 
1 Democritus contended that some worlds were without animals 
and plants because the requisite fluid was lacking which should 
supply them with nourishment. And this dictum of the sage 
is especially remarkable inasmuch as it was obviously based on 
the assumption of the uniformity of the universe in the substances 
composing it and in the laws controlling it, which the sidereal 
physics of our own day has proved beyond dispute. He evinced 
the same spirit which animated Metrodorus of Chios, himself a 
Democritean, in his brilliant parable: "a single ear of corn on 
a wide-spreading champaign would not be more wonderful than 



ch.xiii LEUCIPPUS AND DEMOCRITUS 127 

a single cosmos in the infinitude of space." The genius of Democritus 
did not stop at anticipating modern cosmology.' . . . 

This is a fascinating picture, but surely it is, in any case, much 
overdrawn. And, even if it were true, we cannot but ask, why is 
Anaxagoras, who, before Democritus, spoke of other worlds than 
ours, with their suns and moons, their earths inhabited by men and 
animals, where there are cities and cultivated fields, 'as with us', 
given none of the credit for a theory which ' deposed in principle the 
geocentric hypothesis'? Anaxagoras clearly set no limit to the 
number of such worlds, and Democritus added little to his statement 
except the details that at any given time some of the infinite number 
of worlds are coming into being, others waxing, others waning, others 
being destroyed, and that they represent all possible varieties of 
composition (some with suns and moons, some without, &c), instead 
of being more or less on the same plan with ours, as Anaxagoras 
perhaps implied. Again, the abandonment of the geocentric hypo- 
thesis does not carry us a step towards the Copernican theory 
unless some other and truer centre is substituted for the earth. 
But Democritus's theory of the infinity of worlds does not suggest 
any such centre, nay, it destroys the possibility of there being such 
a centre at all. 1 

With regard, however, to our sun and moon, Democritus puts 
forward a rather remarkable hypothesis connected with the infinite 
multiplication of his worlds. With Anaxagoras the stars, and 
presumably the sun and moon also, were stones torn from the 
earth by the whirling motion of the universe, and afterwards 
kindled into fire by the rapidity of that motion. But according to 
Democritus the sun and moon, which at the time of their coming 
into being * had not yet completely acquired the heat characteristic 
of them, still less their great brilliance, but on the contrary were 
assimilated to the nature subsisting in the earth ' were then ' moving 
in independent courses of their own (κατ ιδίαν) ' ; 'for each of the 
two bodies, when it first came into being, was still in the nature of 
a separate foundation or nucleus for a world, but afterwards, as the 
circle about the sun became larger, the fire was caught up in it \ 2 

1 Cf. Aristotle's argument {De caelo i. 6, 275 b 13) that the universe cannot 
be infinite because the infinite cannot have a centre. 

2 Ps. Plut. Stromat. (apud Euseb. Pr. Ev. i. 8. 7) ; D. G. p. 581 ; Vors. i 2 , 
P• 359• 47. 



128 THE ATOMISTS parti 

The last words appear to relate only to the addition of fire to the 
earthy nucleus of the sun, which may be connected with the idea of 
Leucippus that ' the sun was kindled by the stars ' : but it seems to 
be implied by the whole passage that the sun and the moon, after 
beginning to come into being as the nucleus of separate worlds, were 
caught up by the masses moving round the earth and then carried 
round the earth with them so as to form part of our universe. 

As regards the planets, we have seen that Anaxagoras, like 
Plato, placed the moon nearest to the earth, the sun further from 
it, and the planets further still ; Democritus made the order, 
reckoning from the earth, to be Moon, Venus, Sun, the other 
planets, the fixed stars. 1 ' Even the planets have not all the same 
height ' (i.e. are not at the same distance from us). 2 Seneca 
observes that ' Democritus, the cleverest of all the ancients, says 
he suspects that there are several stars which have a motion of 
their own, but he has neither stated their number nor their names, 
the courses of the five planets not having been at that time under- 
stood \ 3 This seems to imply that Democritus did not even 
venture to say how many planets there were ; Zeller, however, 
holds that he could not but have known of the five planets, 
especially as he wrote a book c about the planets' ; 4 it may be that 
he said in this work that there might perhaps be more planets than 
the five generally known, and Seneca, who had this at third hand, 
may have misunderstood the observation. 5 

An interesting remark about Democritus's views on the motion of 
the sun and moon is contained in a passage of Lucretius, 6 where the 
question is raised, why the sun takes a year to describe the full 
circle of the zodiac while the moon completes its course in a month ; 
perhaps, says Lucretius, Democritus may be right when he says that 
the nearer any body is to the earth, the less swiftly can it be carried 
round by the revolution of the heaven ; now the moon is nearer 
than the sun, and the sun than the signs of the zodiac ; therefore 
the moon seems to get round faster than the sun because, while the 
sun, being lower and therefore slower than the signs, is left behind 

1 Aet. ii. 15. 3 (D. G. p. 344; Vors. i 2 , p. 366. 32). 

2 Hippol. Refut. i. 13. 4 (D. G. p. 565 ; Vors. i 2 , p. 360. 17). 

3 Seneca, Nat. Quaest. vii. 3. 2 (Vors. i 2 , p. 367. 29). 

4 Thrasyllus ap. Diog. L. ix. 46 (Vors. i 2 , p. 357. 22). 

6 Zeller, i 5 , p. 896 note. 6 Lucretius, v. 621 sqq. 



ch.xiii LEUCIPPUS AND DEMOCRITUS 129 

by them, the moon, being still lower and therefore slower still, 
is still more left behind. Therefore it is the moon which appears 
to come back to every sign more quickly than the sun does, be- 
cause the signs go more quickly back to her. The view that the 
bodies which move round at the greatest distance move the most 
quickly and vice versa is the same as we find attributed by Alexan- 
der Aphrodisiensis to the Pythagoreans. 1 

Lastly, we are told by Censorinus 2 that Democritus put the 
Great Year at * 82 years with the same number, 28, of intercalary 
months ', where the c same number ' is the number of intercalary 
months assumed by Callippus in his cycle of 76 years. Tannery 3 
conjectures that the reading should be 77 years (lxxvii) instead of 
82 years (lxxxii), which seems probable enough ; but, as he says, 
it is impossible to draw any certain conclusion from the passage. 

1 Alexander, In metaphysica A. 5, p. 542 a 16-18 Brandis, p. 40. 7-9 Hayduck. 

2 Censorinus, De die natali 18. 8 {Vors. i 2 , p. 390. 19). 

3 Tannery in Mem. de la Socidtides sciences phys. et nat. de Bordeaux, 3 Θ ser. 
iv, 1888, p. 92. 



1410 Κ 



XIV 
OENOPIDES 

The date of Oenopides of Chios is fairly determined by the 
statement of Proclus that he was a little younger than Anaxagoras. 1 
He was a geometer of some note; Eudemus credited him with 
having been the first to investigate the problem of Eucl. I. 13 (the 
drawing of a perpendicular to a given straight line from a given 
point outside it), which he 'thought useful for astronomy', and 
to discover the problem solved in Eucl. I. 23 (the construction on 
a given straight line and at a point on it of an angle equal to a 
given rectilineal angle). No doubt perpendiculars had previously 
been drawn by means of some mechanical device such as a set 
square, and Oenopides was the first to give the theoretical con- 
struction as we find it in Euclid ; and in like manner he probably 
discovered, not the problem of Eucl. I. 23 itself, but the particular 
solution of it given by Euclid. 

In astronomy he is said to have made two discoveries of impor- 
tance. The first is that of the obliquity of the ecliptic. It is true 
that Aetius says that both Thales and Pythagoras, as well as the 
successors of the latter, distinguished the oblique circle of the zodiac 
as touching or meeting three of the ' five circles which are called 
zones ' ; 2 Aetius further states that ' Pythagoras is said to have 
been the first to observe the obliquity of the zodiac circle, a fact 
which Oenopides put forward as his own discovery '. 3 Now Thales 
could not possibly have known anything of the zones, and no doubt 
1 Pythagoras and his successors' may have been substituted for ' the 
Pythagoreans' in accordance with the usual tendency to attribute 
everything to the Master himself; in like manner the second 

1 Proclus, Co?nm. on Eucl. I, p. 66. 2 ( Vors. i 2 , p. 229. 36). 

2 Aet. ii. 12. 1 (D. G. p. 340). 

* Aet. ii. 12. 2 {D. G. p. 340-1 ; Vors. i 2 , p. 230. 14). 



OENOPIDES 131 

passage is probably the result of the same jealousy for the reputa- 
tion of Pythagoras. And for the attribution of this particular 
discovery to Oenopides we have the better authority of Eudemus 
in a passage taken from Dercyllides by Theon of Smyrna. 1 
Macrobius observes that Apollo (meaning the sun) is called Loxias, 
as Oenopides says, because he traverses the oblique circle (λοξον 
κνκλον), moving from west to east. 2 The Egyptian priests, we 
are told, claimed that it was from them that Oenopides learned 
that the sun moves in an inclined orbit and in a sense opposite 
to that of the motion of the other stars. 3 It does not appear that 
Oenopides made any measurement of the obliquity ; at all events 
he cannot be credited with the estimate of 24°, which held its own 
till the time of Eratosthenes (circa 275-194 B. C.). 4 

1 Theon of Smyrna, p. 198. 14, Hiller ( Vors. i 2 , p. 230. 11). 

2 Macrobius, Sat. i. 17. 31 (Vors. i 2 , p. 230. 22). 

3 Diodorus Siculus, i. 98. 2 (Vors. i 2 , p. 230. 19). 

4 Dercyllides' quotation from Eudemus (Theon of Smyrna, pp. 198, 199), 
which states that Oenopides was the first to discover the obliquity of the zodiac 
circle, also mentions that it was other astronomers not named in the particular 
passage who added (among other things) the discovery that the measure of the 
obliquity was the angle subtended at the centre of a circle by the side of a 
regular fifteen-angled figure inscribed in the circle, that is to say, 24 . But this 
value was discovered before Euclid's time, for Proclus, quite credibly, mentions 
(Co?nm. on Eucl. I, p. 269. 11-21) that the proposition Eucl. IV. 16, showing how 
to describe a regular fifteen-angled figure in a circle, was inserted in view of its 
use in astronomy. The value was doubtless known to Eudoxus also, if it does 
not even go back to the Pythagoreans. The angle might no doubt have been 
calculated by means of Pytheas's measurement of the midday height of the sun 
at Marseilles at the summer solstice. According to Strabo (ii. 5. 8, p. 115, and 
ii. 5. 41, p. 134, Cas.), Pytheas found that the ratio of the gnomon to its midday 
shadow at the summer solstice at Marseilles was 120: 41 £ (Ptolemy made it 
6o:2o|, or 120:41$, Syntaxis, ii. 6, p. no. 5). But we are not told of any 
value that Pytheas gave for the latitude of Massalia. According to Strabo, 
Hipparchus said that the same ratio of the gnomon to the shadow as Pytheas 
found at Massalia held good at Byzantium also, whence, relying on Pytheas's 
accuracy, he inferred that the two places were on the same parallel of latitude. 
As, however, Marseilles is 2° further north than Byzantium, it is clear that there 
must have been an appreciable error of calculation somewhere. Theon of 
Alexandria (On Ptolemy's Syntaxis, p. 60) states that Eratosthenes discovered 
the distance between the tropic circles to be n/83rds of the whole meridian 
circle = 47 42' 40", which gives 23 5 1 ' 20" for the obliquity of the ecliptic. Berger, 
however (Die geographischen Fragmente des Eratosthenes, 1880, p. 1 31), is 
inclined to infer from Ptolemy's language that it was Ptolemy himself who 
invented the ratio n : 83, and that Eratosthenes still adhered to the value 24 . 
For Ptolemy (Syntaxis i. 12, p. 67. 22-68. 6) says that he himself found the 
distance between the tropic circles to lie always between 47 40' and 47°45', 
* from which we obtain about (σχ&όν) the same ratio as that of Eratosthenes, 
which Hipparchus also used. For the distance between the tropics becomes 
(or is found to be, -yiWai) very nearly 1 1 parts out of 83 contained in the whole 

Κ % 



4 



132 



OENOPIDES part ι 



The second discovery attributed to Oenopides is that of a Great 
Year, the duration of which he put at 59 years. 1 In addition, we 
are told by Censorinus that Oenopides made the length of the year 
to be 365!^ days. 2 Tannery 3 suggests the following as the method 
by which he arrived at these figures. Starting first of all with 
365 days as the length of a year, and 29J days as the length of the 
lunar month, approximate values known before his time, Oenopides 
had to find the least integral number of complete years which 
would contain an exact number of lunar months ; this is clearly 
59 years, which contains a number of lunar months represented by 
twice $6$, or 730. He had then to determine how many days 
there were in 730 months. This his knowledge of the calendar 
would doubtless enable him to do, and he would appear to have 
arrived at 21,557 days as the result, 4 since this, when divided by 59, 
gives 365I-I days as the length of the year. 5 Tannery gives good 

meridian circle.' The mean between 47°4o' and 47 45' is of course 47 42' 30", 
or only 10" different from 47°42'4o"; but the wording is somewhat curious if 
Ptolemy meant to imply that the actual ratio II 183 represented Eratosthenes' 
estimate. For ' the same ratio ' would then be 1 1/83 and σχεδόν and Ζγγιστα 
would have to mean exactly the same thing. Moreover, in that case, to make 
a separate sentence of the comparison with the fraction n/83 was quite un- 
necessary ; all that was necessary was to add to the preceding sentence some 
words such as ' namely 1 i/83rds of the meridian circle ' in explanation of ' the 
same ratio'. On the other hand, if the intention was to compare the mean 
value 47 42' 30" with a value 48 , or 2/i5ths of a great circle, used by Eratos- 
thenes and Hipparchus, there was a sort of excuse for a separate sentence 
converting 47 42' 30" into a fraction of a great circle as nearly as possible equi- 
valent, namely I i/83rds, for the purpose of comparison with 2/i5ths, the difference 
between the fractions being 1/1245. Hipparchus, in his Commentary on the 
Phaenomena of Aratus and Eudoxus (p. 96. 20-21, Manitius) said that the 
summer tropical circle is ' very nearly 24 north of the equator'. Another value for 
the obliquity of the ecliptic is derivable from an obiter dictum of Pappus (vi. 35, 
p. 546. 22-7, ed. Hultsch). Pappus, without any indication of his source, there 
says that the value of the ratio which we should call the tangent of the angle is 
10/23. We should scarcely have expected a ratio between such small numbers 
to give a very accurate value, but 10/23 = 0-4347826, which is the tangent of an 
angle of 23 29' 55" nearly. 

1 Theon of Smyrna, p. 198. 15 (Vors. i 2 , p. 230. 13) : Aelian, V. H. x. 7 {Vors. 
i 2 , p. 230. 27) ; Aet. ii. 32. 2 {D. G. p. 363 ; Vors. i 2 , p. 230. 34). 

2 Censorinus, De die natali 19. 2. 

3 Tannery in Mem, de la Societi des sciences phys. et nat. de Bordeaux, 
3 Θ ser. iv. 1888, pp. 90, 91. 

4 The true synodic month being 29-53059 days, 730 times this gives, as a 
matter of fact, 21557J days nearly. 

5 This year of a little less than 365 days 9 hours is slightly more correct than 
the average year of the octa'eteris of 2923^ days, which works out to 365 days 
10J hours (Ginzel, Handbuch der mathe?natischen und technischen Chronologie> 
vol. ii, 191 1, p. 387). 



ch.xiv OENOPIDES 133 

ground for thinking that Oenopides cannot have taken account of 
the motion of all the planets as well as of the sun and moon for 
the purpose of calculating the Great Year. He would, no doubt, 
know the approximate periods of revolution of Saturn, Jupiter, 
and Mars, namely 30 years for Saturn, 12 years for Jupiter, and 
2 years for Mars, which figures would give roughly, in his great 
year of 59 years, 2 revolutions of Saturn, 5 of Jupiter, and 30 or 
31 for Mars. Admitting the last number as the more exact, and 
dividing 21,557 days by these numbers respectively, we obtain 
periods for the revolution of the several planets which, like the 
figures worked out by Schiaparelli for Philolaus, would show errors 
not exceeding 1 per cent, of the true values. But Tannery considers 
that this is not the proper way to judge of the error ; he would 
rather judge the degree of inaccuracy by the error in the mean 
position of the planet at the end of the period. He finds that, 
calculated on this basis, the error would not reach as much as 
2° in the case of Saturn, and 9 in the case of the sun ; but for 
Mars the error would exceed 107 , which is quite inadmissible. 
If Oenopides had ventured to indicate the sign of the zodiac in 
which each planet would be found at the end of his period, the 
error in the case of Mars would have been discovered when the 
time came. 

Aristotle 1 says that some of the so-called Pythagoreans held 
that the sun at one time moved in the Milky Way. This same 
view is attributed to Oenopides ; for Achilles says 2 that ' According 
to others, among whom is Oenopides of Chios, the sun formerly 
moved through this region [the Milky Way], but because of the 
Thyestes-feast he was diverted and has (since) revolved in a path 
directed the opposite way to the other, that namely which is 
defined by the zodiac circle \ 

1 Aristotle, Meteorologica i. 8, 345 a 16 ( Vors. i 2 , p. 230. 39). 

2 Achilles, hagoge ad Arat. 24, p. 55. 18, Maass (Vors. i 2 , p. 230. 42). 



XV 
PLATO 

In order to obtain an accurate view of Plato's astronomical system 
as a whole, and to judge of the value of his contributions to the 
advance of scientific astronomy, it is necessary, first, to collect and 
compare the various passages in his dialogues in which astronomical 
facts or theories are stated or indirectly alluded to ; then, secondly, 
allowance has to be made for the elements of myth, romance, and 
idealism which are, in a greater or less degree depending on the 
character of the particular dialogue, invariably found as a setting 
and embellishment of actual facts and theories. When these ele- 
ments are as far as possible eliminated, we find a tolerably com- 
plete and coherent system which, in spite of slight differences of 
detail and a certain development and even change of view between 
the earlier and the later dialogues, remains essentially the same. 

In considering this system we have further to take into account 
Plato's own view of astronomy as a science. This is clearly stated 
in Book VII of the Republic, where he is describing the curriculum 
which he deems necessary for training the philosophers who are to 
rule his State. The studies required are such as will lift up the 
soul from Becoming to Being ; they should therefore have nothing 
to do with the objects of sensation, the changeable, the perishable, 
which are the domain of opinion only and not of knowledge. It is 
true that sensible objects are useful in so far as they give the 
stimulus to the purely intellectual discipline required, in so far, in 
fact, as they suffice to show that sensations are untrustworthy or 
even self-contradictory. Some objects of perception are adequately 
appreciated by the perception ; these are non-stimulants ; others 
arouse the intellect by showing that the mere perception produces 
an unsound result. Thus the perception which reports that a thing 
is hard frequently reports that it is also soft, and similarly with 



PLATO 135 

thickness and thinness, greatness and smallness, and the like. In 
such cases the soul is perplexed and appeals to the intellect for 
help ; the intellect responds and looks at 'great' and ' small' (e.g.) 
as distinct and not confounded ; we are thus led to the question 
what is the ' great ' and what is the ' small \ Science then is only 
concerned with realities independent of sense-perception ; sensation, 
observation, and experiment are entirely excluded from it. At the 
beginning of the formulation of the curriculum for philosophers 
gymnastic and music are first mentioned, only to be rejected at 
once; gymnastic has to do with the growth and waste of bodies, 
that is, with the changeable and perishing ; music is only the 
counterpart, as it were, of gymnastic. Next, all the useful arts are 
tabooed as degrading. The first subject of the curriculum is then 
taken, namely the science of Number, in its two branches οι αριθμη- 
τική, dealing with the Theory of Numbers, as we say, and of 
λογιστική, calculation, with the proviso that it is to be pursued for 
the sake of knowledge and not for purposes of trade. Next comes 
geometry, and here Plato, carrying his argument to its logical con- 
clusion, points out that the true science of geometry is, in its nature, 
directly opposed to the language which, for want of better terms, 
geometers are obliged to use ; thus they speak of ' squaring ', 
' applying ' (a rectangle), ' adding ', &c, as if the object were to do 
something, whereas the true purpose of geometry is knowledge. 
Geometrical knowledge is knowledge of that which is, not of that 
which becomes something at one moment and then perishes ; and, 
as such, geometry draws the soul towards truth and creates the 
philosophic spirit which helps to raise up what we wrongly keep 
down. Astronomy is next mentioned, but Socrates corrects him- 
self and gives the third place in the curriculum to stereometry, or 
solid geometry as we say, which, adding a third dimension, 
naturally follows plane geometry. And fourth in the natural order 
is astronomy, since it deals with the ' motion of body ' (φορά 
βάθους, literally ' motion of depth ' or of the third dimension). 

When astronomy was first mentioned, Socrates' interlocutor 
hastened to express approval of its inclusion, because it is proper, 
not only for the agriculturist and the sailor, but also for the general, 
to have an adequate knowledge of seasons, months, and years ; 
whereupon Socrates rallies him upon his obvious anxiety lest the 



I3 6 PLATO part ι 

philosopher should be thought to be pursuing useless studies. 
When the speakers return to astronomy after the digression on 
solid geometry, Glaucon tries a different tack : at all events, he says, 
astronomy compels the soul to look upward and away from the 
things of the earth. But no ! he is using the term ' upward ' in the 
sense of towards the material heaven, not, as Socrates had meant 
it, towards the realm of ideas or truth ; and Socrates at once takes 
him up. On the contrary, he says, as it is now taught by those 
who would lead us upward to philosophy, it is calculated to turn 
the soul's eye down. 

* You seem with sublime self-confidence to have formed your 
own conception of the nature of the learning which deals with the 
things above. At that rate, if a person were to throw his head 
back and learn something by contemplating a carved ceiling, you 
would probably suppose him to be investigating it, not with his 
eyes, but with his mind. You may be right, and I may be wrong. 
But I, for my part, cannot think any other study to be one that 
makes the soul look upwards except that which is concerned with 
the real and the invisible, and, if any one attempts to learn anything 
that is perceivable, I do not care whether he looks upwards with 
mouth gaping or downwards with mouth closed : he will never, as 
I hold, learn — because no object of sense admits of knowledge — 
and I maintain that in that case his soul is not looking upwards but 
downwards, even though the learner float face upwards on land or 
in the sea/ * I stand corrected,' said he ; ' your rebuke was just. 
But what is the way, different from the present method, in which 
astronomy should be studied for the purposes we have in view ?' 

'This', said I, 'is what I mean. Yonder broideries in the 
heavens, forasmuch as they are broidered on a visible ground, are 
properly considered to be more beautiful and perfect than anything 
else that is visible ; yet they are far inferior to those which are true, 
far inferior to the movements wherewith essential speed and essen- 
tial slowness, in true number and in all true forms, move in relation 
to one another and cause that which is essentially in them to move : 
the true objects which are apprehended by reason and intelligence, 
not by sight. Or do you think otherwise?' 'Not at all,' said he. 
' Then ', said I, ' we should use the broidery in the heaven as illus- 
trations to facilitate the study which aims at those higher objects, 
just as we might employ, if we fell in with them, diagrams drawn 
and elaborated with exceptional skill by Daedalus or any other 
artist or draughtsman ; for I take it that any one acquainted with 
geometry who saw such diagrams would indeed think them most 



ch. xv PLATO 137 

beautifully finished but would regard it as ridiculous to study them 
seriously in the hope of gathering from them true relations of 
equality, doubleness, or any other ratio.' ' Yes, of course it would 
be ridiculous/ he said. ' Then ', said I, c do you not suppose that 
one who is a true astronomer will have the same feeling when he 
looks at the movements of the stars ? That is, will he not regard 
the maker of the heavens as having constructed them and all that 
is in them with the utmost beauty of which such works admit ; yet, 
in the matter of the proportion which the night bears to the day, 
both these to the month, the month to the year, and the other stars 
to the sun and moon and to one another, will he not, think you, 
regard as absurd the man who supposes these things, which are 
corporeal and visible, to be changeless and subject to no aberrations 
of any kind ; and will he not hold it absurd to exhaust every 
possible effort to apprehend their true condition ?' * Yes, I for one 
certainly think so, now that I hear you state it.' ' Hence ', said I, 
1 we shall pursue astronomy, as we do geometry, by means of pro- 
blems, and we shall dispense with the starry heavens, if we propose 
to obtain a real knowledge of astronomy, and by that means to 
convert the natural intelligence of the soul from a useless to a use- 
ful possession.' ' The plan which you prescribe is certainly far more 
laborious than the present mode of studying astronomy/ l 

We have here, expressed in his own words, Plato's point of view, 
and it is sufficiently remarkable, not to say startling. We follow 
him easily in his account of arithmetic and geometry as abstract 
sciences concerned, not with material things, but with mathematical 
numbers, mathematical points, lines, triangles, squares, &c, as 
objects of pure thought. If we use diagrams in geometry, it is only 
as illustrations ; the triangle which we draw is an imperfect repre- 
sentation of the real triangle of which we think. And in the 
passage about the inconsistency between theoretic geometry and 
the processes of squaring, adding, &c, we seem to hear an echo 
of the general objection which Plato is said to have taken to the 
mechanical constructions used by Archytas, Eudoxus, and others 
for the duplication of the cube, on the ground that ' the good of 
geometry is thereby lost and destroyed, as it is brought back to 
things of sense instead of being directed upward and grasping at 
eternal and incorporeal images \ 2 But surely, one would say, the 
case would be different with astronomy, a science dealing with 

1 Republic vii. 529 A-530 B. 

2 Plutarch, Quaest. Conviv. viii. 2. I, p. 718 F (Vors. i 2 , p. 255. 3-5). 



138 PLATO parti 

the movements of the heavenly bodies which we see. Not at all, 
says Plato with a fine audacity, we do not attain to the real science 
of astronomy until we have l dispensed with the starry heavens \ 
i.e. eliminated the visible appearances altogether. The passage 
above translated is admirably elucidated by Dr. Adam in his edition 
of the Republic} There is no doubt that Plato distinguishes two 
astronomies, the apparent and the real, the apparent being related 
to the real in exactly the same way as practical (apparent) geometry 
which works with diagrams is related to the real geometry. On the 
one side there are the visible broideries or spangles in the visible 
heavens, their visible movements and speeds, the orbits which they 
are seen to describe, and the number of hours, days, or months 
which they take to describe them. But these are only illustrations 
(παραδείγματα) of real heavens, real spangles, real or essential speed 
or slowness, real or true orbits, and periods which are not days, 
months, or years, but absolute numbers. The broideries or span- 
gles in both the astronomies are stars, but stars regarded as moving 
bodies. Essential speed and essential slowness seem to be, as Adam 
says, simply mathematical counterparts of visible stars, because they 
are said to be carried in the true motions of real astronomy, and 
therefore cannot be the speed and slowness of the mathematical 
bodies of which the visible stars are illustrations, but must be those 
mathematical bodies themselves. The true figures in which they 
move are their mathematical orbits, which we might now say are 
the perfect ellipses of which the orbits of the visible material planets 
are imperfect copies. And lastly, as a visible planet carries with it 
all the sensible properties and phenomena which it exhibits, so does 
its mathematical counterpart carry with it the mathematical realities 
which are in it. In short, Plato conceives the subject-matter of 
astronomy to be a mathematical heaven of which the visible heaven 
is a blurred and imperfect expression in time and space ; and the 
science is a kind of ideal kinematics, a study in which* the visible 
movements of the heavenly bodies are only useful as illustrations. 

But, we may ask, what form would astronomical investigations 
on Plato's lines have taken in actual operation ? Upon this there 
is naturally some difference of opinion. One view is that of 

1 See, especially, vol. ii, pp. 128-31, notes, and Appendices II and X to Book 
VII, pp. 166-8,186-7. 



ch. xv PLATO 139 

Bosanquet, 1 who relies upon the phrase ' we shall pursue astronomy 
as we do geometry by means of problems \ and suggests that the 
discovery of Neptune, picturesquely described by De Morgan as 
' Leverrier and Adams calculating an unknown planet into visible 
existence by enormous heaps of algebra \ 2 is the kind of investiga- 
tion which 'seems just to fulfil Plato's anticipations'. Plato was 
a master of method, and it is an attractive hypothesis to picture 
him as having at all events foreshadowed the methods of modern 
astronomy ; but Adam seems to be clearly right in holding that 
the illustration does not fit the language of the passage in the 
Republic which we are discussing. For Plato says that the person 
who thought that the heavenly bodies should always move pre- 
cisely in the same way and show no aberrations whatever would 
properly be thought 'absurd', and that it would be absurd to 
exhaust oneself in efforts to make out the truth about them ; hence, 
on this showing, the visible perturbations of Uranus would scarcely 
have seemed to Plato very extraordinary or worth any very deep 
investigation by 'heaps of algebra' or otherwise. Besides, the 
discovery of Uranus's perturbations could hardly have been made 
without observation, and observation is excluded by the words ■ we 
shall let the heavens alone '. The fact is that, at the time when 
our passage was written, Plato's ' problems ' were a priori problems 
which, when solved, would explain visible phenomena ; Adams 
began at the other end, with observations of the phenomena, and 
then, when these were ascertained, sought for their explanation. 

It may be that, when Plato is banning sense-perception from the 
science of astronomy in this uncompromising manner, he is con- 
sciously exaggerating ; it would not be surprising if his enthusiasm 
and the strength of his imagination led him to press his point 
unduly. In any case, his attitude seems to have changed con- 
siderably by the time when he wrote the Timaeus and the Laws, 
both as regards the use made of sense-perception and the relation 
of astronomy to the visible heaven. In the Republic sense-percep- 
tion is only regarded as useful up to the point at which, owing to 
its presentations contradicting one another, it stimulates the intellect. 
In the Timaeus the senses, e.g. sight, fulfil a much more important 

1 Bosanquet, Companion to Plaid's Republic ', 1 895, pp. 292-3. 

2 De Morgan, Budget of Paradoxes, p. 53. 



140 PLATO part ι 

role. ' Sight, according to my judgement, has been the cause of 
the greatest blessing to us, inasmuch as of our present discourse 
concerning the universe not one word would have been uttered had 
we never seen the stars and the sun and the heavens. But now day 
and night, being seen of us, and months and revolutions of years 
have made number, and they gave us the notion of time and 
the power of searching into the nature of the All ; whence we have 
derived philosophy, than which no greater good has come nor shall 
come hereafter as the gift of the gods to mortal man. This 
I declare to be the chiefest blessing due to the eyes/ 1 In the 
Laws Plato makes the Athenian stranger say that it is impious 
to use the term ' planets ' of the gods in heaven as if they and the 
sun and moon never kept to one uniform course, but wandered 
hither and thither ; the case is absolutely the reverse of this, ' for 
each of these bodies follows one and the same path, not many paths 
but one only, which is a circle, although it appears to be borne in 
many paths.' 2 Here then we no longer have the view that the 
visible heavenly bodies should be neglected as being subject to 
perturbations which it would be useless to attempt to fathom, 
and that true astronomy is only concerned with the true heavenly 
bodies of which they are imperfect copies ; but we are told that 
the paths of the visible sun, moon, and planets are perfectly uniform, 
the only difficulty being to grasp the fact. Bosanquet observes, 
on the passage in the Republic contrasting the visible and the true 
heavens, that ' Plato's point is that there are no doubt true laws by 
which the periods, orbits, accelerations and retardations of the solids 
in motion can be explained, and that it is the function of astronomy 
to ascertain them'. 3 On the later view stated in the Laws this 
would be true with ' the visible heavenly bodies ' substituted for 
' solids in motion '. 

We are told on the authority of Sosigenes, 4 who had it from 
Eudemus, that Plato set it as a problem to all earnest students 
to find ' what are the uniform and ordered movements by the 
assumption of which the apparent movements of the planets can 
be accounted for '. The same passage says that Eudoxus was the 

1 Timaeus 47 α, Β. 2 Laws vii. 822 A. 

3 Bosanquet, Companion, p. 291. 

4 Simplicius on De caelo ii. 12 (292 b 10), p. 488. 20-4, Heib. 



ch.xv PLATO 141 

first to formulate hypotheses with this object ; Heraclides of Pontus 
followed with an entirely new hypothesis. Both were pupils of 
Plato, and it is a fair inference that the stimulus of the Master's 
teaching was a factor contributing to these great advances, although 
it is probable that Eudoxus attacked the problem on his own 
initiative. 

When we come to extract from the different dialogues the details 
of Plato's astronomical system, we find, as already indicated, that, 
if allowance is made for the differences in the literary form in which 
they are presented, and for the greater or less admixture of myth, 
romance, and poetry, the successive presentations of the system 
at different periods of Plato's life merely show different stages of 
development ; the system remains throughout fundamentally the 
same. Some of the passages have nothing mythical about them 
at all ; e.g. the passage in the Laws, which is intended to combat 
prevailing errors, gives a plain statement of the view which Plato 
thought the most correct. In the passages in which myth has 
a greater or less share, that which constitutes the most serious part 
is precisely that which relates to astronomy ; and that which proves 
that the astronomical part is serious is the fact that, in different 
forms, and with more or fewer details in different passages, we have 
only one and the same main hypothesis ; the variations are on 
points which are merely accessory. 1 Nor was the system revolu- 
tionary as compared with previous theories ; on the contrary, Plato 
evidently selected what appeared to him to be the best of the 
astronomical theories current in his time, and only made corrections 
which his inexorable logic and his scientific habit of mind could not 
but show to be necessary ; and the theory which commended itself 
to him the most was that of Pythagoras and the early Pythagoreans 
— the system in which the earth was at rest in the centre of the 
universe — as distinct from that of the later Pythagorean school, with 
whom the earth became a planet revolving like the others about the 
central fire. 

Plato's system is set out in its most complete form in the 
Timaeus, and on this ground Martin, in his last published memoir 
on the subject, began with the exposition in the Timaeus and then 

1 Cf. Martin in Memoires de Γ Acad, des Inscriptions et Belles-Lettres, xxx, 
1881, pp. 6-13. 



1 4 3 PLATO PARTI 

added, for the purpose of comparison, the substance of the astrono- 
mical passages in the other dialogues. This plan would perhaps 
enable a certain amount of repetition to be avoided ; but I think 
that the development of the system is followed better if the usual 
plan is adopted and the dialogues taken in chronological order. 

We begin therefore with the Phaedrus, perhaps the earliest of all 
the dialogues. The astronomy in the Phaedrus consists only in the 
astronomical setting of the myth about souls soaring in the heaven 
and then again falling to earth. Soaring in the heaven, they with 
difficulty keep up for a time with the chariots of the gods in their 
course round the heavens. 

* Zeus, the great captain in heaven, mounted on his winged 
chariot, goes first and disposes and oversees all things. Him follows 
the army of Gods and Daemons ordered in eleven divisions ; for 
Hestia alone abides in the House of God, while, among the other 
gods, those who are of the number of the twelve and are appointed 
to command lead the divisions to which they were severally 
appointed. 

Many glorious sights are there of the courses in the heaven 
traversed by the race of blessed gods, as each goes about his own 
business ; and whosoever wills, and is able, follows, for envy has no 
place among the Heavenly Choir . . . 

The chariots of the gods move evenly and, being always obedient 
to the hand of the charioteer, travel easily ; the others travel with 
great difficulty . . . 

The Souls which are called immortal, when they are come to the 
summit of the Heaven, go outside and stand on the roof and, as 
they stand, they are carried round by its revolution and behold 
the things which are outside the Heaven.' 1 

Here, then, the army of Heaven is divided into twelve divisions. 
One is commanded by Zeus, the supreme God, who also commands- 
in-chief all the other divisions as well ; subject to this, each division 
has its own commander. Zeus is here the sphere of the fixed stars, 
which revolves daily from east to west and carries round with it 
the other divisions except one, Hestia, which abides unmoved in the 
middle. Hestia, the Hearth in God's House, stays at home to 
keep house; the other divisions follow the march of Zeus but 
perform separate evolutions under the command of their several 
leaders. Hestia is here undoubtedly the earth, unmoved in the 

1 Plato, Phaedrus 246 E-247 C. 



ch. xv PLATO 143 

centre of the world, 1 and is not the central fire of the Pythagoreans. 
The gods in command of the ten other divisions are, in the first 
place, the seven planets, i.e. the sun and moon and the five planets, 
and then between them and the earth come the three others which 
are the aether, the air, and the moist or water. 2 The sun, moon, 
and planets are all carried round in the general revolution of the 
whole heaven from east to west, but have independent duties and 
commands of their own, i.e. separate movements which (as later 
dialogues will tell us) are movements in the opposite sense, i. e. from 
west to east. 

In the Phaedo Plato puts into the mouth of Socrates his views as 
to the shape of the earth, its position and its equilibrium in the 
middle of the universe. The first passage on the subject is that in 
which he complains of the inadequate use by Anaxagoras of his 
Notes in explaining phenomena. 

4 When once I heard some one reading from a book, as he said, of 
Anaxagoras, in which the author asserts that it is Mind which dis- 
poses and causes all things, I was pleased with this cause, as it seemed 
to me right in a certain way that Mind should be the cause of all 
things, and I thought that, if this is so, and Mind disposes everything, 
it must place each thing as is best. . . . With these considerations in 
view I was glad to think that I had found a guide entirely to my 
mind in this matter of the cause of existing things, I mean Anaxa- 
goras, and that he would first tell me whether the earth is flat or 
round, and, when he had told me this, would add to it an explana- 
tion of the cause and the necessity for it, which would be the Better, 
that is to say, that it is better that the earth should be as it is ; and 
further, if he should assert that it is in the centre, that he would 
add, as an explanation, that it is better that it should be in the 
centre. . . . Similarly I was prepared to be told in like manner, 
with regard to the sun, the moon, and the other stars, their relative 
speeds, their turnings or changes, and their other conditions, in what 
way it is best for each of them to exist, to act, and to be acted upon 
so far as they are acted upon. For I should never have supposed 
that, when once he had said that these things were ordered by Mind, 
he would have assigned to them in addition any cause except the 
fact that it is best that they should be as they are. . . . From what 

1 Cf. Theon of Smyrna, p. 200. 7 ; Plutarch, De ftrt7no frigido, c. 21, p. 954 F ; 
Proclus, In Titnaewn, p. 281 Ε ; Chalcidius, Timaeus, c. 122, p. 187, and 
c. 178, pp. 227-8. 

2 Chalcidius, loc. cit. ; cf. Proclus, In retnfi. vol. ii, p. 130, 6-9, Kroll. 



144 PLATO parti 

a height of hope then was I hurled down when I went on with my 
reading and saw a man that made no use of Mind for ordering things, 
but assigned as their cause airs, aethers, waters, and any number of 
other absurdities.' [Then follows the sentence stating that it is as if 
one were to say that Socrates did everything he did by Mind and 
then gave as the cause of his sitting there the fact that his body was 
composed of bones and sinews, the former having joints, and the 
sinews serving to bend and stretch out the limbs consisting of 
the bones with their covering sinews and flesh and skin, and so on. 
This inability to distinguish between what is the cause of that 
which is and the indispensable conditions without which the cause 
cannot be a cause suggests that most people are fumbling in the 
dark.] ' Thus it is that one makes the earth remain stationary 
under the heaven by making it the middle of a vortex, another sets 
the air as a support to the earth, which is like a flat kneading- 
trough.' x 

The last sentence alludes to some of the familiar early views as to 
the form of the earth. Only Parmenides and the Pythagoreans 
thought it to be spherical, the Ionians and others supposed it to be 
flat, though differing as to details ; the theory that it is the motion 
of a vortex with the earth in the middle that keeps it stationary is 
that of Empedocles, while the idea that it is a disc, or like a flat 
kneading-trough, supported by air, is of course that which Aristotle 
attributes to Anaximenes, Anaxagoras, and Democritus. 2 

Plato's own view is stated later in the dialogue. 

1 There are many and wondrous regions in the earth, and it is 
neither in its nature nor in its size what it is supposed to be by those 
whom we commonly hear speak about it ; of this I have been con- 
vinced, I will not say by whom. . . . My persuasion as to the form of 
the earth and the regions within it I need not hesitate to tell you . . . 
I am convinced then, said he, that, in the first place, if the earth, 
being a sphere, is in the middle of the heaven, it has no need either 
of air or of any other such force to keep it from falling, but that the 
uniformity of the substance of the heaven in all its parts and 
the equilibrium of the earth itself suffice to hold it ; for a thing in 
equilibrium in the middle of any uniform substance will not have 
cause to incline more or less in any direction, but will remain as 
it is, without such inclination. In the first place I am persuaded 
of this.' 3 

1 Phaedo 97B-99B. 2 Aristotle, De caelo ii. 13, 294 b 13. 

3 Phaedo 108C-IC9A. 



ch. xv PLATO 145 

When Socrates says he has been convinced by some one of the 
fact that the earth is different from what it was usually supposed to 
be, he is considered by some to be referring to Anaximander who 
drew the first map of the inhabited earth. But surely Anaxi- 
mander's views, no doubt with improvements, would be represented 
in those of his successors, the geographers of the time, whom 
Socrates considers to be wrong (we are told, for instance, that 
Democritus, who, like Anaximander, thought the earth flat, com- 
piled a geographical and nautical survey of the earth 1 ). ' Some one ' 
may possibly be no one in particular, in accordance with Plato's 
habit of ' giving an air of antiquity to his fables by referring them to 
some supposititious author '. 2 On the other hand, the explanation of 
the reason why the spherical earth remains in equilibrium in the 
centre of the universe, namely that there is nothing to make it move 
one way rather than another, is sufficiently like Anaximander's 
explanation of the same thing. 3 

Socrates proceeds : 

* Moreover, I am convinced that the earth is very great, and that 
we who live from the river Phasis as far as the Pillars of Heracles 
inhabit a small part of it ; like to ants or frogs round a pool, so we 
dwell round the sea ; while there are many other men dwelling 
elsewhere in many regions of the same kind. For everywhere on 
the earth's surface there are many hollows of all kinds both as 
regards shapes and sizes, into which water, clouds, and air flow and 
are gathered together ; but the earth itself abides pure in the purity 
of the heaven, in which are the stars, the heaven which the most 
part of those who use to speak of these things call aether, and it is 
the sediment of the aether which, in the forms we mentioned, is 
always flowing and being gathered together in the hollow places 
of the earth. We then, dwelling in the hollow parts of it, are not 
aware of the fact but imagine that we dwell above on its surface ; 
this is just as if any one dwelling down at the bottom of the sea 
were to imagine that he dwelt on its surface and, beholding the sun 
and the other heavenly bodies through the water, were to suppose 
the sea to be the heaven, for the reason that, through being sluggish 
and weak, he had never yet risen to the top of the sea nor been able, 
by putting forth his head and coming up out of the sea into the 
place where we live, to see how much purer and more beautiful it is 

1 Agathemerus, i. I (Vors. i 2 , p. 393. 6, 7). 

2 Archer-Hind, The Phaedo of Plato, p. 161 note. 

3 See pp. 24, 25, above : cf. Aristotle, De caelo ii. 13, 295 b 11. 
i«o L 



146 PLATO part ι 

than his abode, neither had heard this from another who had seen 
it. We are in the same case ; for, though dwelling in a hollow of 
the earth, we think we dwell upon its surface, and we call the air 
heaven as though this were the heaven and through this the stars 
moved, whereas in fact we are through weakness and sluggishness 
unable to pass through and reach the limit of the air ; for, if any- 
one could reach the top of it or could get wings and fly up, then, 
just as fishes here, when they come up out of the sea, espy the 
things here, so he, having come up, would likewise descry the things 
there, and if his strength could endure the sight would know that 
there is the true heaven, the true light, and the true earth. For 
here the earth, with its stones and the whole place where we are, is 
corrupted and eaten away, as things in the sea are eaten away by 
the salt, insomuch that there grows in the sea nothing of moment 
nor anything perfect, so to speak, but there are hollow rocks, sand, 
clay without end, and sloughs of mire wherever there is also earth, 
things not worthy at all to be compared to the beautiful objects 
within our view ; but the things beyond would appear to surpass 
even more the things here.' l . . . 

Then begins the myth of the things which are upon the real 
earth and under the heaven. 

1 First it is said that, if one saw it from above, the earth is like 
unto a ball made with twelve stripes of different colours, each stripe 
having its own colour. . . .' 

We need not pursue the picture of the idealized earth with its 
varied hues, its precious stones, its race of men excelling us in sight, 
hearing, and intelligence in the same proportion as air excels water, 
and aether excels air, in purity, and so on. 

Reading the story of the hollows in the earth, we recall the idea 
of Archelaus, which he perhaps learnt from Anaxagoras, that the 
earth was hollowed out in the middle but higher at the edges. This 
shape would correspond to the flat kneading-trough mentioned by 
Plato as the form given by some to the earth. 2 Plato, realizing 
that certain inhabited regions such as that from the river Phasis 
(descending from the Caucasus into the Black Sea) to the Pillars of 
Hercules, being partly bounded by mountains, did appear to be 
hollows, had to reconcile this fact with his earnest conviction of the 
earth's sphericity. Archelaus regarded the whole earth as one such 

1 Phaedo 109A-HOA. 2 Ibid. 99 B. 



ch. xv PLATO 147 

hollow ; to which Plato replies that the inhabited earth may be a 
hollow, but it is not the whole earth. The earth itself is very large 
indeed, so that the apparent hollow formed by the portion in which 
we live is quite a small portion of the whole. There are any number 
of other hollows of all sorts and sizes ; these hollows are separated 
by the ridges between them, and it is only the tops of these ridges 
that are on the real surface of the spherical earth. Consequently 
there is nothing in the existence of the hollows that is inconsistent 
with the earth being spherical ; they are mere indentations. The 
impossibility of our climbing up the sides to the top of the bounding 
ridges, or taking wings and flying out of the hollows, and so reach- 
ing the real surface of the earth and obtaining a view of the real 
heavens, is of course poetic fancy and has nothing to do with 
astronomy. 

The extreme estimate of the size of the earth made by Plato in 
the Phaedo seems to be peculiar to him. For the sake of contrast, 
Aristotle's remarks on the same subject may be referred to. 1 Aris- 
totle says that observations of the stars show not only that the 
earth is spherical, but that it is ' not great \ For quite a small 
change of position from north to south or vice versa involves a 
change of the circle of the horizon. Thus some stars are seen in 
Egypt and Cyprus which are not seen in the northern regions, and 
some stars which in the northern regions are always above the 
horizon are, in Egypt, seen to rise and set. Such differences for so 
small a change in the position of an observer would not be possible 
unless the earth's sphere were of quite moderate size. Aristotle 
adds that the mathematicians of his day who tried to calculate the 
circumference of the earth made it approach 400,000 stades. This 
estimate had, according to Archimedes, 2 been reduced in his time 
to 300,000 stades, and Eratosthenes made the circumference to be 
252,000 stades on the basis of a definite measurement of the arc 
separating Syene and Alexandria on the same meridian, compared 
with the known distance between those places. 

On the negligibility of the height of the highest mountain in 
comparison with the diameter of the earth, Theon of Smyrna 3 has 

1 Aristotle, De caelo ii. 14, 297 b 30-298 a 20. 

2 Archimedes, Sand-reckoner (vol. ii, ed. Heib., p. 246. 15 ; ed. Heath, p. 222). 
8 Theon of Smyrna, pp. 124-6, Hiller. 

L 2 



1 48 PLATO PART I 

some remarks based on the estimates of 252,000 stades for the 
circumference and of 10 stades (a low estimate, it is true) for 
the height of the highest mountain above the general level of the 
plains. 

S Coming now to the Republic^ Book X, we get a glimpse of a 
more complete system, though again the astronomy is blended with 
myth. The story is that of Er, the son of Armenius, who, after 
being killed in battle, came to life twelve days afterwards and 
recounted what he had seen. He first came with other souls to 
a mysterious place where there were two pairs of mouths, one pair 
leading up into heaven, the other two down into the earth ; between 
them sat judges who directed the righteous to take the road to the 
right hand leading up into the heaven and sent those who had 
wrought evil down the left-hand road into the earth ; at the same 
time other souls were returning by the other road out of the earth, 
and others again by the other road coming down from the heaven : 
the two returning streams met, the former travel-stained after 
a thousand years' journeying under the earth, the latter returning 
pure from heaven, and they foregathered in the meadow where 
they related their several experiences. 

' Now when seven days had passed since the spirits arrived in the 
meadow, they were compelled to arise on the eighth day and 
journey thence ; and on the fourth day they arrived at a point from 
which they saw extended from above through the whole heaven and 
earth a straight light, like a pillar, most like to the rainbow, but 
brighter and purer. This light they reached when they had gone 
forward a day's journey, and there, at the middle of the light, they 
saw, extended from heaven, the extremities of the chains thereof ; 
for this light it is which binds the heaven together, holding together 
the whole revolving firmament as the undergirths hold together 
triremes ; and from the extremities they saw extended the Spindle 
of Necessity by which all the revolutions are kept up. The shaft 
and hook thereof are made of adamant, and the whorl is partly 
of adamant and partly of other substances. 

Now the whorl is after this fashion. Its shape is like that we use ; 
but from what he said we must conceive of it as if we had one great 
whorl, hollow and scooped out through and through, into which was 
inserted another whorl of the same kind but smaller, nicely fitting 
it, like those boxes which fit into one another ; and into this again 
we must suppose a third whorl fitted, into this a fourth, and after 
that four more. For the whorls are altogether eight in number, set 



ch. xv PLATO 149 

one within another, showing their rims above as circles and forming 
about the shaft a continuous surface as of one whorl ; while the 
shaft is driven right through the middle of the eighth whorl. 

The first and outermost whorl has the circle of its rim the 
broadest, that of the sixth is second in breadth, that of the fourth 
is third, that of the eighth is fourth, that of the seventh is fifth, 
that of the fifth is sixth, that of the third is seventh, and that of the 
second is eighth. And the circle of the greatest is of many colours, 
that of the seventh is brightest, that of the eighth has its colour 
from the seventh which shines upon it, that of the second and fifth 
are like each other and yellower than those aforesaid, the third 
is the whitest in colour, the fourth is pale red, and the sixth is the 
second in whiteness. 

The Spindle turns round as a whole with one motion, and within 
the whole as it revolves the seven inner circles revolve slowly in the 
opposite sense to the whole, and of these the eighth goes the most 
swiftly, second in speed and all together go the seventh and sixth 
and fifth, third in the speed of its counter-revolution the fourth 
appears to move, fourth in speed comes the third, and fifth the 
second. And the whole Spindle turns in the lap of Necessity. 

Upon each of its circles above stands a Siren, carried round with 
it and uttering one single sound, one single note, and out of all the 
notes, eight in number, is formed one harmony. 

And again, round about, sit three others at equal distances apart, 
each on a throne, the daughters of Necessity, the Fates, clothed 
in white raiment and with garlands on their heads, Lachesis, 
Clotho, and Atropos, and they chant to the harmony of the Sirens, 
Lachesis the things that have been, Clotho the things that are, and 
Atropos the things that shall be. 

And Clotho at intervals with her right hand takes hold of the 
outer revolving whorl of the Spindle and helps to turn it ; Atropos 
with her left hand does the same to the inner whorls ; Lachesis 
with both hands takes hold of the outer and inner alternately 
(i.e. of the outer with her right hand and of the inner with 
her left).' l 

On the precise interpretation of the details of this description 
there has been a great deal of discussion and difference of opinion. 3 
Some of the details are hardly astronomical, and this is not the 
place for more than a short statement of the principal points at 
issue. 

1 Republic tl. 616 B- 61 7 D. 

2 Very full information will be found in Adam's edition of the Republic ; see 
especially the notes in vol. ii, pp. 441-53, and Appendix VI to Book X, 
pp. 470-9• 



J 5° 



PLATO part ι 



First, what is the form and position of the ' straight light, like 
a pillar ', and at what point is ' the middle ' of the light where the 
souls saw 'the extremities of the chains' binding the heavens 
together? As early as Proclus's time one supposition was that 
the light was the Milky Way. 1 Proclus rejected this view, which 
in modern times is represented by Boeckh 2 and Martin. 3 Boeckh 
supposes the souls to be beyond the north pole, outside the circle 
of the Milky Way which, if seen from the outside edgeways, would 
look straight ; the middle of the light is for him the north pole, 
from which stretch the chains of heaven, one of which is the 
lio-ht. Martin makes the souls see the Milky Way as a straight 
column of light from below ; thence they go quickly up in the day's 
journey to the middle of the light (Martin compares the souls in 
Phaedrus 247 B-348 B, which get to the outside of the sphere of 
the fixed stars) ; they there see both poles of the sphere, and the 
curved column is, for them, like a band forming a complete ring 
round the sphere and holding it together ; this curved column can 
only be the Milky Way. Martin supports his view by pressing 
the comparison of the column to a rainbow, which, he says, must 
refer to its form and not to its colours; and for the illusion of 
supposing the curved column to be straight he cites the parallel 
of Xenophanes, who thought the stars moved in straight lines 
which only appeared to be circles. I agree with Adam's opinion 
that to suppose the column to be curved and only to appear 
straight does violence to the language of Plato. Then again, it 
would be strange that the souls, one class of which has come back 
from a thousand years' journey in the heaven, and the other from 
the same length of journey under the earth, should next be taken 
up, all of them, to the top of the heavenly sphere ; there is nothing 
to suggest that, either in the four days elapsing between the time 
when they leave the meadow and the time when they first see the 
straight column of light, or in the one day following which brings 
them to the middle of the light, they leave the earth at all. The 
other alternative is to take the ' straight light ' to be, in accordance 
with the natural meaning of the words, a straight line or straight 

1 Proclus, In remp. vol. ii, p. 194. 19, Kroll. 

2 Boeckh, Kleine Schriften, in, pp. 266-320. 

3 Martin in Μ dm. de Ρ Acad, des Inscriptions et Belles-Letires, xxx, 1881, 
pp. 94-7. 



ch.xv PLATO 151 

cylindrical column of light passing from pole to pole right through 
the centre of the universe and of the earth (occupying the centre 
of the universe), which column of light symbolizes the axis on 
which the sphere of the heaven revolves. Where then is ' the 
middle' of this column of light which the souls are supposed to 
reach one day after they first see the column ? Adam thinks 
it can only be at the centre of the earth, and he seems to base 
this view mainly on the fact that, later on, the souls, after passing 
under the throne of Necessity and encamping by the river of 
Unmindfulness in the plain of Lethe, are said (621 B) to go up, 
'shooting like stars/ to be born again. Here also I cannot but 
think it strange that all the souls should be brought down to the 
centre of the earth, seeing that one class of them had just returned 
from a thousand years' wandering in the interior of the earth, to 
say nothing of the shortness of the time allowed for reaching the 
centre of the earth, namely, one day from the time when they first 
saw the column of light, while there is nothing in the language 
describing the five days' journey to suggest that they did anything 
but walk (πορζνζσθαή. Now the place of the judgement-seat which 
was between the mouths of the earth and the heaven, and to 
which the souls returned after their thousand years in the earth and 
heaven respectively, was on the surface of the earth ; presumably 
therefore the meadow to which they turned aside from that place 
was also on the surface of the earth (and not even on the surface 
of the ' True Earth ' of the PAaedo, as Adam supposes) ; and 
Mr. J. A. Stewart * has pointed out that the popular belief as to 
the river Lethe made it a river entirely above ground and not one 
of the rivers of Tartarus. Hence I am disposed to agree with 
Mr. Stewart that the whole journey from the meadow by the 
throne of Necessity to the plain of the river Lethe was along the 
surface of the earth. Although Adam rightly rejects Boeckh's 
identification of the 'straight light' with the Milky Way, he is 
induced by the parallel of the ' undergirths ' (ύποζώματα) of 
triremes to assume, in addition to the straight light forming the 
axis of the universe, a circular ring of light passing round it from 
pole to pole and joining the straight portion at the poles ; 2 this 

1 J. A. Stewart, The Myths of Plato, pp. 154 sqq. 

2 Adam, The Republic of Plato, vol. ii, pp. 445—7, notes. 



15α PLATO part ι 

he does because the more proper meaning of ' undergirths ' appears 
to be ropes passed round the vessel outside it and horizontally, 
rather than planks passing longitudinally from stem to stern as 
Proclus and others supposed. 1 But there is nothing in the Greek 
to suggest the addition of this circle to the straight light ; and 
the assumption seems, as Mr. Stewart says, 2 to make too much 
of the man-of-war or trireme. Moreover, the ground for assuming 
a ring, as well as a straight line, of light vanishes altogether if 
the νποζώματα are, after all, cables stretched tight, i.e. in straight 
lines, inside the ship from stem to stern, as Tannery holds. 3 It 
seems to be enough to regard Plato as saying that the pillar (which 
alone is mentioned) holds the universe together in its particular 
way as the undergirths do the trireme in their way. I prefer then 
to believe that the light is simply a straight column or cylinder 
of light, and that the c middle of the light ' is the point on the 
surface of the earth which is in the centre of the column of light, 
i.e. the centre of the circular projection of the cylinder of light on 
the earth's surface. I do not see why the souls, looking from that 
point along the cylinder of light in both directions, should not in 
this way be supposed to see (illuminated by the column as by a 
searchlight) the poles of the universe, nor why these should not 
be called the extremities of the chains holding the heaven together, 
the pillar of light having by a sudden change of imagery become 
those chains themselves. 

The Spindle of Necessity, 

By another sudden change of imagery the chains following the 
course of the pillar of light become a spindle which is similarly 
extended from the same ' extremities ' or poles, and the spindle 
with its whorls representing the movements of the universe is seen 
to turn in the lap of Necessity. The throne of Necessity must on 

1 Proclus, In retnfi. vol. ii, p. 200. 25, and scholium, ibid. p. 381. 10. 

2 Stewart, op. cit., p. 169. 

3 Tannery in Revue de Philologie, xix, 1895, p. 117 : 'Le Thesaurus constate, 
d'ailleurs, que Boeckh a demontre que les νποζώματα vt&v, dont il est assez 
souvent fait mention dans les inscriptions, sont des cables, ainsi que du reste 
Hesychius [s.v. ζωμςύματα] explique ce mot : σχοινιά κατά μίσον την νανν δ(σμ€υό- 

μίνα. Ces cables e*taient tendus, d'apres les Origines a'lsiaort, entre l'etrave et 
l'e'tambot, en tout cas, on ne peut se les figurer tendus autrement que suivant 
une ligne droite.' 



ch.xv PLATO 153 

the above view be at the point on the surface of the earth which 
is in the middle of the column of light ; and on this hypothesis, as 
on others, the attempt to translate the details of the poetic imagery 
into a self-consistent picture of physical facts is hopeless, for the 
simple reason that one thing cannot both be entirely outside 
another thing and entirely within it at the same time. Let us 
assume with Boeckh that the souls are outside the universe when 
they see the apparently straight light ; Necessity will then pre- 
sumably be outside the universe which in the form of the spindle 
and whorls she holds in her lap. It is on this assumption im- 
possible to give an intelligible meaning to ' under the throne of 
Necessity ' as an intermediate point on the journey of the souls 
from the meadow to the plain of Lethe. The same difficulty 
arises if, with Zeller, we suppose Plato to be availing himself of 
the external Necessity which, according to Aetius, Pythagoras 
regarded as * surrounding the world ', 1 Plato's Necessity is cer- 
tainly not outside but in the middle. If, however, Necessity 
sits either at the centre of the earth as supposed by Adam, or at 
a point on the surface of the earth as supposed by Mr. Stewart, 
how can she, being inside the universe, hold the spindle and whorls 
forming the universe in her lap? This is no doubt the difficulty 
which makes Mr. Stewart infer that Necessity does not hold the 
universe itself in her lap, but a model of the universe. 2 

The whorls. 

The real astronomy of the Republic is contained in the description 
of the whorls and their movements. The first question arising is, 
what was the shape of the whorls ? They are not spheres because 
they have rims ('lips', χ^ίλη) one inside the other, which are all 
visible and form one continuous flat surface as of one whorl. We 
might, on the analogy of Parmenides' bands, suppose that they 
are zones of hollow spheres symmetrical about a great circle, i.e. so 
placed that the plane of the great circle is parallel to, and equi- 
distant from, the outer circles bounding the zones. Adam supposes 
them to be hemispheres^ which Plato possibly obtained by cutting 



1 Aet. i. 25. 2 (D. G. p. 321). 

2 Stewart, op. cit., pp. 152-3, 165. 



i54 PLATO parti 

in half the Pythagorean spheres mentioned by Theon of Smyrna. 1 
It is true that there is nothing in the text of Plato requiring them 
to be hemispheres, although Proclus regards them as segments of 
spheres 2 ; but the supposition that they are hemispheres has the 
great advantage that it eliminates all question of the depth of 
the whorls measured perpendicularly (downwards, let us say) from 
the visible flat surface formed by their rims. Plato says nothing 
of the depth of the whorls, but merely gives the rims different 
breadths. The moment we suppose the whorls to be zones or rings 
we have to consider what depth or thickness (i.e. perpendicular 
distance between the two bounding surfaces) must be assigned to 
them. The thickness of the rings would presumably be great 
enough to hold symmetrically the largest of the heavenly bodies 
which the rings carry round with them. 3 Martin 4 takes the 
thickness of the rings to be greater than this ; he supposes that 
the outer whorl is an equatorial zone of the celestial sphere 
included between two equal circular sections 'which are doubtless 
the tropics'. But Martin admits that there is, in the whole passage, 
no reference to any obliquity of movements relatively to the 
equator, and he can only suppose such obliquity to be tacitly implied 
by the thickness of each whorl. I think that this supposition is 
unsafe, and that it is better to assume that, at this stage in the 
development of his astronomy, or perhaps merely for the purpose 
of the imagery of this particular myth, Plato did not recognize 
any obliquity, still less any variations of obliquity in the movements 
of the planets. 5 I prefer therefore to suppose the whorls to be 

1 Theon of Smyrna, p. 150. 14. 

2 Proclus, In remp. vol. ii, p. 213. 19-22. 

3 The revolving whorls π^ριάγονσι τους αστέρας (Proclus, In remp. vol. ii, 
p. 226, 12). 

4 Martin in Mem. de VAcademie des Inscriptions et Belles-Lettres, xxx, 1881, 
pp. IOO-I. 

6 Yet Berger {Geschichte der wissenschaftlichen Erdkunde der Griechen, 1903, 
pp. 199-201) still insists on regarding Plato's ' breadths ' as what I have called 
depths. According to him a 'lip ' (χβίλο?) must project (cf. Plato, Critias 
115 E) ; hence he thinks they must project and recede in comparison with one 
another. It is difficult, as he sees, to reconcile this with νωτον συνεχές, ' a con- 
tinuous back ' as seen from above, say the pole ; he is therefore driven to the 
supposition that the words may describe the appearance of the outermost whorl 
as seen from a position where it hides all the others, i.e. from a point between 
the planes of its bounding circles ; but this clearly will not do. The object of 
Berger is to make out that Plato wished to distinguish by the ' breadths ' of his 
rings the inclinations of the movements of the several planets. As I have said 



ch.xv PLATO 155 

hemispheres, or similar segments of spheres fitting one inside the 
other, and having their bases in one plane. The planets, sun, and 
moon would perhaps be regarded as fixed in such a position that 
their centres would be on the plane surface which is the common 
boundary of all the whorls, so that half of each planet would 
project above that surface and half of it would be below. 

It is not difficult to see what is the astronomical equivalent of 
each of the concentric whorls. The outermost (the first) represents 
the sphere of the fixed stars ; and here we have somewhat the same 
difficulty as we saw in the case of Parmenides' wreaths or bands. 
The fixed stars being spread over the whole sphere, how can that 
sphere be represented by a hemisphere, or a segment of a sphere, or 
a ring or zone ? The answer is presumably that the whorls are 
pure mechanism, designed with reference to the necessity of making 
the movements of the inner whorls give plane circular orbits to the 
seven single heavenly bodies, the sun, the moon, and the five planets. 
Mr. Stewart, in accordance with his idea that it is a model which 
Necessity holds in her lap, suggests that the model might be an 
old-fashioned one with rings instead of spheres, or that, if it were 
an up-to-date model, with spheres, it might be one in which only the 
half of each sphere was represented so that the internal 'works' 
might be seen ; he compares the passage in the Timaens l where the 
speaker says that, without the aid of a model of the heavens, it 
would be useless to attempt to describe certain motions. 

above, there is nothing in the text to suggest any obliquity in the movements ; 
and, if the i breadths ' are depths, the sizes of the rings as measured by their 
inner and outer radii become entirely indeterminate, so that the relative orbital 
distances are undistinguished. It is quite incredible that Plato should say 
nothing about the relative sizes of the orbits while carefully distinguishing their 
obliquities relatively to the equator. It is true that Aristotle, Metaph. Λ. 8, 
1073 b 17 sq.) and Theon of Smyrna (p. 174. 1-3) admit different obliquities 
exhibited by the planetary motions ; and Cleomedes {De motte circulari ii. 7, 
p. 226. 9-14) gives some estimates of them. These are, however, all obliquities 
with reference to the ecliptic, not the equator. Moreover, Cleomedes' figures 
are quite irreconcilable with Plato's corresponding ' breadths '. Cleomedes says 
that the obliquity is the greatest in the case of the moon ; next comes Venus 
which diverges 5 on each side of the zodiac ; next Mercury, 4 ; next Mars 
and Jupiter, 2\° each ; and last of all Saturn, i°. Plato places them in 
descending order of ' breadth ' thus : Venus, Mars, Moon, Mercury, Jupiter, 
Saturn. 

1 Timaeus 40 D. Cf. Theon of Smyrna, p. 146. 4, where Theon alludes to 
the same passage of the Timaeus, and says that he himself made a model 
to represent the system described in the present passage of the Republic, 



1^6 PLATO part ι 

The second whorl (reckoning from the outside) carries the planet 
Saturn, the third Jupiter, the fourth Mars, the fifth Mercury, the 
sixth Venus, the seventh the sun, and the eighth the moon. The 
earth, as always in Plato, is at rest in the centre of the system. 
The outer rim of each whorl clearly represents the path of the 
heavenly body which that whorl carries. The breadth of each 
whorl, that is, the difference between the radii of its outer and inner 
rims respectively (the inner radius of the particular whorl being of 
course the outer radius of the next smaller whorl), is the difference 
between the distances from the earth of the planet carried by the 
particular whorl and of the planet carried by the next smaller 
whorl. The rim of the innermost whorl (the eighth) is the orbit of 
the moon, the outer rim of the next whorl (the seventh) is the orbit 
of the sun, and so on. Proclus * says that there was an earlier 
reading of the passage about the breadths of the rims of the 
successive whorls which made them dependent on, i.e. presumably 
proportional to, the sizes of the successive planets. Professor Cook 
Wilson observes that ' this principle would be a sort of equable 
distribution of planetary mass, allowing the greater body more 
space. It would come to allowing the same average of linear 
dimension of planetary mass to each unit of distance between orbits 
throughout the system.' 2 Adam, however, for reasons which he 
gives, decides in favour of our reading of the passage as against the 
' earlier ' reading of Proclus. 

As regards the speeds we are told that, while the outermost whorl 
(the sphere of the fixed stars) and the whole universe (including 
the inner whorls) along with it are carried round in one motion of 
rotation in one direction (i.e. from east to west), the seven inner 
whorls have slow rotations of their own in addition, the seven 
rotations being at different speeds but all in the opposite sense to 
the rotation of the whole universe. Hence the quickest rotation is 
that of the fixed stars and the whole universe, which takes place 
once in about 24 hours ; the slower speeds of the rest are speeds 
which are not absolute but relative to the sphere of the fixed stars 
regarded as stationary, and of these relative speeds the quickest is 
that of the moon, the next quickest that of the sun, Venus, and 

1 Proclus, In remp. vol. ii, p. 218, 1 sq. Cf. Theon of Smyrna, p. 143. 14-16. 

2 See Adam, Plato's Republic^ vol. ii, pp. 475-9. 



ch.xv PLATO 157 

Mercury, which travel in company with one another, i.e. have the 
same angular velocity and take about a year to describe their orbits 
respectively ; the next is that of Mars, the next that of Jupiter, and 
the last and slowest relative motion is that of Saturn. The speeds 
here are all angular speeds because, if the sun, Venus, and Mercury 
describe their several orbits in the same time, the sun must have the 
least linear velocity of the three, Venus the next greater, and Mercury 
the greatest, since the actual length of the orbit of the sun is less 
than that of the orbit of Venus, and the length of the orbit of Venus 
is again less than that of the orbit of Mercury. To obtain the 
absolute angular speeds in the direction of the daily rotation, 
i. e. from east to west, we have to deduct from the speed of the 
daily rotation the slower relative speeds of the respective planets 
in the opposite sense ; the absolute angular speeds are therefore, in 
descending order, as follows : 

Mercury' 
Sphere of fixed stars, Saturn, Jupiter, Mars, - 



Venus 
Sun 



Moon. 



The following table gives the order of orbital distances, or 
breadths of rims of whorls, as compared with the order of the 
whorls themselves, the order of relative speeds, and the relation of 
the colours of the planets respectively : 

q j . Order in 

jt/j , τι, . breadth of rim ,.* . , .. * Relation of 

Whorl. Planet. r- . according, to relative 7 J 

according to n 7 , ή 7 ,, , , colours. 

. . ϊ• Proclus s old speeds, 

our reading. ,. r 
ώ reading. 

1 — Spangled. 

7 5 Yellower than 
sun and moon. 

6 4 Whitest. 

5 3 Rather red. 

8 ii \ Like Saturn in 
colour. 

Second in white- 
ness. 
Brightest. 
3 I Light borrowed 

from sun. 



1 = 

2 = 


Sphere of fixed stars 
Saturn 


1 

8 


3 = 

4 = 

5 = 


Jupiter 

Mars 

Mercury 


7 
3 
6 


6 = 


Venus 


2 


7 = 

8 = 


Sun 
Moon 


5 
4 



2 v 2 



As, according to either reading, Plato only gives the order of the 
successive rims as regards breadth, not the ratios of their breadths, 
we cannot gather from this passage what was his view as to the 



158 PLATO part ι 

ratios of the distances of the respective heavenly bodies from 
the earth. Nor can his estimate of the ratios be deduced from the 
mere allusion to the harmony produced by the eight notes chanted 
by the Sirens perched upon the respective whorls ; as to this har- 
mony see pp. 105-15 above. 

As regards the Sirens, Theon of Smyrna tells us that some sup- 
posed them to be the planets themselves ; some, however, regarded 
them as representing the several notes which were produced by the 
motion of the several stars at their different speeds. 1 It is clear 
that the latter is the right view ; the Sirens are a poetical expression 
of the notes. 

It will be noticed that Plato has the correct theory with regard 
to the moon's light being derived from the sun, a fact which, as 
before stated, he evidently learned from Anaxagoras. 

The Titnaens is one of the latest of Plato's dialogues and is the 
most important of all for our purpose because in it Plato's astro- 
nomical system is most fully developed and given with the fewest 
lacunae. I shall continue to follow the plan of quoting passages in 
Plato's own words and adding the explanations which appear 
necessary. First, we are told that the universe is one only, eternal, 
alive, perfect in all its parts, and in shape a perfect sphere, 2 that 
being the most perfect of all figures. 

* He (the Creator) assigned it that motion which was proper to its 
bodily form, that motion of all the seven which most belongs to 
reason and intelligence. Wherefore turning it about uniformly, in 
the same place, and in itself, he made it to revolve round and 
round; but all the other six motions he took away from it and 
stablished it without part in their wanderings.' 3 

'And in the midst of it he put soul and spread it throughout the 
whole, and also wrapped the body with the same soul round about 
on the outside ; and he made it a revolving sphere, a universe one 
iind alone.' 4 

Here then we have all plurality of worlds denied and the one 
universe made to revolve uniformly, carrying with it in its revolution 
all that is within it, as in the Republic ; the uniform revolution is of 
course the daily rotation. Turning ' in itself means about its own 
axis and therefore, so to speak, coincidently with itself, so that one 

1 Theon of Smyrna, pp. 146. 8-147. 6. 2 Timaeus 32 c-33 B. 

3 Ibid. 34 A. 4 Timaeus 34 B. 



ch. xv PLATO 159 

position does not overlap another, but in all positions the sphere 
occupies exactly the same space and place. The other ' six motions ' 
from which it is entirely free are the three pairs of translatory 
motions, forward and backward, right and left, up and down. 

Next Plato explains how the Creator made the Soul by first 
combining in one mixture Same, Other, and Essence, and then 
ordering the mixture according to the intervals of a musical scale, 
so that its harmony pervaded the whole substance. This substance, 
considered as having taken the form of a bar or band, a soul-strip 
as it were, he proceeds to divide. 

* Next he cleft the structure so formed lengthwise into two halves 
and, laying them across one another, middle upon middle in the 
shape of the letter X, he bent them in a circle and joined them, 
making them meet themselves and each other at a point opposite 
to that of their original contact ; and he comprehended them in that 
motion which revolves uniformly and in the same place, and one of 
the circles he made exterior and one interior. The exterior move- 
ment he named the movement of the Same, the interior the 
movement of the Other. The revolution of the circle of the Same 
he made to follow the side (of a rectangle) towards the right hand, 
that of the circle of the Other he made to follow the diagonal and 
towards the left hand, and he gave the mastery to the revolution of 
the Same and uniform, for he left that single and undivided ; but 
the inner circle he cleft, by six divisions, into seven unequal circles 
in the proportion severally of the double and triple intervals, each 
being three in number; and he appointed that the circles should 
move in opposite senses, three at the same speed, and the other 
four differing in speed from the three and among themselves, yet 
moving in a due ratio.' 1 

The two circles in two planes forming an angle and bisecting one 
another at the extremities of a diameter common to both circles 
are of course the equator and the zodiac or ecliptic. The equator 
is the circle of the Same, the ecliptic that of the Other. In the 
accompanying figure, AEBF is the circle of the Same (the equator), 
CEDE the circle of the Other (the ecliptic), and they intersect at 
the ends of their common diameter EF. GH is the axis of the 
universe which is at right angles to the plane of the circle AEBF . 
If we draw chords DK y CL parallel to the diameter AB common to 
the circles AEBF, AGBH, and join CK y DL, we have a rectangle 

1 Timaeus 36 B-D. 



τ 6ο 



PLATO 



PART I 



of which KD is a side and CD is a diagonal. As the universe 
revolves round GH 9 each point on the circumference of the circle 
AG Β Η describes a circle parallel to the circle AEBF, i.e. a 

circle about a diameter parallel 
to A Β or KD ; that is, the revo- 
lution ' follows the side ' KD of 
the rectangle. Similarly the 
revolution of the circle of the 
Other about an axis perpen- 
dicular to the plane of the circle 
CFDE ' follows the diagonal ' 
CD of the rectangle. 

The circle of the Same or the 
equator is the outer, and the circle 
of the Other, the ecliptic, is the 
inner. When Plato says that the 




Fig. 4. 



Creator Comprehended them ' (i.e. both circles) in the motion of 
the Same, and then again later that he gave the supremacy to that 
circle, he means that the movement of that circle is common to the 
whole heaven and carries with it in its motion the smaller circles, 
the subdivisions of the circle of the Other, and everything in the 
universe ; this he makes still clearer in a later passage where he 
speaks of the motion of the planets in the circle of the Other being 
1 controlled ' by the motion of the Same, and the motion of the 
Same twisting all their circles into spirals. 1 The subjection of all 
that is in the universe, including all the independent motions of the 
planets, to the one general movement of daily rotation is of course 
the same as we saw in the Republic ; but there all the circles were 
in one plane, whereas the bodies moving in the opposite sense to 
the daily rotation here move in a different plane, that of the ecliptic, 
instead of that of the equator. 

I have represented the directions of the motions in the two circles 
by arrows in the figure. The motion in the circle AEBF is in the 
direction represented by the order of the letters. 

The statement of Plato that the Creator made the circle of the 
Same (i.e. the circle of the fixed stars) revolve towards the right 
hand and the circle of the Other (comprising the circles of the 

1 Ttmaeus 39 A. 



ch.xv PLATO ιβτ 

planets) towards the left hand has given the commentators, from 
Proclus downwards, much trouble to explain. It is also in con- 
tradiction to the observation in the Laws that motion to the right 
is motion towards the east, 1 while the writer of the Epinomis again 
represents the independent movement of the sun, moon, and planets 
as being to the right and not to the left. 2 There is of course no 
difficulty in the circumstance that Plato has previously said that 
the Creator took away from the world -sphere the £ix motions, up 
and down, right and left, forwards and backwards ; for this refers 
to movements of translation such as take place inside the sphere, not 
to the revolution of the sphere itself. The axis of such revolution 
being once fixed, the revolution may be in one of two (and only 
two) directions ; 3 consequently there is nothing to prevent one of 
the two directions being described as to the right and the other 
as to the left. But why did Plato speak of the revolution from 
east to west as being motion to the right ? Boeckh has discussed 
the question at great length, giving a full account of earlier views 
before stating his own.* Martin's explanation is that Plato is 
speaking from the point of view of a spectator looking south, as 
he would have to do in northern latitudes in order to see the 
apparent revolution of the sun from east to west ; that is, the 
movement is from left to right. Boeckh, however, points out that 
the Greeks were accustomed, from the earliest times when diviners 
foretold events by watching the flight of birds, to turn their faces 
to the north ; the east would therefore be on the right hand and 
would naturally be regarded as the most auspicious, and therefore 
as ' right '. It is also true that the common View among the Greeks 
(we find it later in Aristotle 5 ) would make of the sphere of the 
universe a sort of world-animal, which would have a right and left 
of its own, as it might be a man masked in a sphere put over him ; 
and no doubt, on such a view, the east would be sure to be 
regarded as c right ' and the west as ' left \ Boeckh therefore finds 
it difficult to believe that Plato could have represented the east 
as left. Assuming then that Plato regarded the east as right, 

1 Laws vi. 8, 760 D. 

2 Epinomis 987 B. 

! The sphere has, in mathematical language, only ' one degree of freedom \ 
* Boeckh, Das kosmische System des Platon, pp. 28-32. 
5 Aristotle, De caelo ii. 2, 285 b 2-3. 

1410 Μ 



i62 PLATO parti 

Boeckh thinks Martin's view untenable, and concludes that the only 
possible alternative is to suppose that Plato must have thought, 
in the Timaeus, of a movement from the right to the right again, 
i.e. of the whole revolution from east to east instead of the portion 
from the east to the west. But the movement, on the assumptions 
made, is undoubtedly left-wise, and it seems to me that Boeckh's 
explanation is almost as violent as the desperate method of inter- 
pretation suggested by Proclus. 1 Where Boeckh is in error is, 
I think, in supposing that Plato would identify the east in his 
world-sphere with the right hand at all ; it seems to me that he 
could not possibly have done so consistently with the scientific 
attitude he adopted in denying the existence of any absolute up 
and down, right and left, forward and backward in the spherical 
universe. He explains, for example, that ' up ' and ' down ' have 
only a relative meaning as applied to different parts of the sphere, 2 
and it is clear that, in the same connexion, he would say the same 
of right and left. Now suppose that a particular point on the 
equator of the universe is east at a given moment ; after about 
six hours the same point will be south, after six more west, and 
so on. The case then is similar to that put by Plato when he says 
that a man going round the circumference of a solid body placed 
at the centre of the universe would at some time arrive at the 
antipodes of an earlier position and would therefore, on the usual 
view of ' up ' and down, have to call ' down ' what he had before 
described as c up ', and vice versa? Plato would never, surely, 
have made the same mistake in speaking of the universe. On 
the contrary, when he spoke of the daily rotation, he properly 
ignored all question of a starting-point, whether east or west, right 
or left, or of the position of a person setting the sphere in motion, 
and confined himself to distinguishing by different names the two 
possible directions of motion in order to make it clear that the 
circles of the Same and of the Other moved in opposite directions. 
The expressions to the right and to the left were obviously well 

1 Proclus (In Timaeum 220 e) will have it that em Se|ta does not mean the 
same thing as els to Se£ioi/, but that, while els to 8e£iov refers to motion in a 
straight line, erci δβ£ιά only refers to motion in a circle and means ' the place to 
which the right moves (anything),' εφ' d το δεξών Kivel. 

2 Plato, Timaeus 62 D-63 E. 
8 Timaeus 62 E-63 A. 



ch. xv PLATO 163 

adapted to express the distinction, and it seems to me that the 
reason of Plato's particular application of them is simply this. He 
considered that the circle of the Same must have the superior 
motion ; but right is superior to left ; he therefore described the 
revolution of the circle of the Same as being to the right, and 
the revolution of the circle of the Other as being to the left, for 
this sole reason, without regard to any other considerations, just 
as in the Republic he confines himself to saying that Clotho at 
intervals, with her right hand, helps to turn the outer whorl of the 
spindle, and so on, 1 without saying anything about the actual 
directions in which the respective whorls revolve. On the other 
hand, when he says in the Laws that revolution from west to east 
is to the right and revolution from east to west is to the left, he 
is, as Boeckh properly observes, merely using popular language. 

The cutting of the circle of the Other into seven concentric 
circles (including the original circumference as one of the seven) 
produces seven orbits in exactly the same way as the eight whorls 
in the Myth of Er give eight orbits, the difference being that the 
outermost circle of the Republic, the circle about which the sphere 
of the fixed stars moves, is not now in the same plane with the 
other seven, but is the circle of the Same in a different plane. 
Plato here says that the seven circles move in opposite directions, 
literally ' in opposite senses to one another ', which, as there are 
only two directions, can only mean that a certain number of the 
seven revolve in one direction, and the rest in the other ; we shall 
return later to this point, which presents great difficulty. The 
three which move at the same speed are of course the circles of the 
sun, Venus, and Mercury, as in the Republic, the same speed 
meaning, as there, not the same linear speed (as they are at 
different distances from the earth), but the same angular speed. 

The seven circles are said to be ' in the proportion of the double 
and triple intervals, three of each\ The allusion is to the 
Pythagorean τετρακτύς represented in the annexed figure, the 
numbers on the one side after 1 being successive powers of 2 
and those on the other side successive powers of 3. When the con- 
centric circles into which the circle of the Other is divided are 
said to correspond to these numbers, it is clear that it must be 

1 Republic x. 617 c, D. 
Μ 2 




1 64 PLATO parti 

the circumferences (or, what is the same thing in other words, the 
radii), not the areas, which so correspond ; for, if it were the areas, 
the radii would not be commensurable with one another. The 
dictum is generally 1 taken to mean that the radii of the successive 
orbits, i.e. the distances between the successive planets and the 
earth, are in the ratio of the numbers i, 2, 3, 4, 8, 9, 27. But 
Chalcidius 2 apparently takes the several numbers to indicate the 
successive differences between radii, for he says that, while the 
first distance (1) is that between the earth and the moon, the second 
(2) is the distance between the moon (not the earth) and the sun ; 
on this view, the successive radii are 1, 1+2 = 3, 
1 +2 + 3 = 6, &c. Macrobius 3 says that the Plato- 
nists made the distances cumulative by way of 
multiplication, the distance of the sun from 
Ή the earth being thus (in terms of the distance of 
ji^ 5 the moon from the earth) ι χ 2 or 2, that of Venus 

1x2x3 = 6, that of Mercury 6 χ 4 = 24, that of 
Mars 24x9 = 216, that of Jupiter 216x8 = 1,728, and that of 
Saturn 1,728x27 = 46,656. (It will be observed that in this 
arrangement 9 comes before 8, Macrobius having previously ex- 
plained this order by saying that, after 1, we first take the first 
even number, 2, then the first odd number, 3, then the second even 
number, 4, then the second odd number, 9, then the third even 
number, 8, and last of all the third odd number, 27.) But, whatever 
the exact meaning, it is obvious that we have here no serious 
estimate of the relative distances of the sun, moon, and planets 
based on empirical data or observations ; the statement is a piece 
of Plato's ideal a priori astronomy, in accordance with his statement 
in the Republic, Book VII, that the true astronomer should 'dis- 
pense with the starry heavens '. 

s Plato goes on to the question of Time and its measurement. As 
the ideal after which the world was created is eternal, but no 
created thing can be eternal, God devised for the world an image 
of abiding eternity ' moving according to number, even that which 
we have named time \ 

1 Cf. Zeller, ii 4 , p. 779 note. 

2 Chalcidius, Timaeus, c. 96, p. 167, ed. Wrobel. 

3 Macrobius, In somn. Scip. ii. 3. 14. 



ch.xv PLATO 165 

' For, whereas days and nights and months and years were not 
before the heaven was created, he then devised their birth along 
with the construction of the heaven. Now these are all portions 
of time. . .' * 

' So, then, this was the plan and intent of God for the birth of 
time ; the sun, the moon, and the five other stars which are called 
planets have been created for defining and preserving the numbers 
of time. 

'And when God had made their several bodies, he set them in 
the orbits in which the revolution of the Other was moving, in 
seven orbits seven stars. The moon he placed in that nearest the 
earth, in the second above the earth he placed the sun ; next, 
the Morning Star and that which is held sacred to Hermes he 
placed in those orbits which move in a circle having equal speed 
with the sun, but have the contrary tendency to it ; hence it is that 
the sun and the star of Hermes and the Morning Star overtake, 
and are in like manner overtaken by, one another. And as to the 
rest, if we were to set forth the orbits in which he placed them, 
and the causes for which he did so, the account, though only by 
the way, would lay on us a heavier task than that which is our 
chief object in giving it. These things, perhaps, may hereafter, 
when we have leisure, find a fitting exposition.' 2 

The crux of this passage is the statement that, while Venus and 

Mercury have the same speed as the sun, i. e. have the same angular 

speed, describing their orbits in about the same time, 'they have 

the contrary tendency to the sun ' ; the words are kvavriav δυναμιν, 

' contrary tendency ' or ' force '. In an earlier passage, as we have 

seen, Plato spoke of some of the seven planets moving on the 

concentric circles forming part of the circle of the Other as going 

'the opposite way' (κατά τάναντία) 3, to the others. Now, although 

δύναμις need not perhaps here be a 'principle of movement' as 

Aristotle defines it, 4 yet if we read the two passages together and 

give the most natural sense to the words in both cases, the meaning 

certainly seems to be that some of the planets describe their orbits 

in the contrary direction to the others, and that those which move, 

in the zodiac, the opposite way to the others are Venus and 

Mercury ; that is to say, the sun, the moon, Mars, Jupiter, and 

Saturn all move in the direction of the motion of the circle of the 

Other, i.e. from west to east, while Venus and Mercury move in 

the same plane of the zodiac but in the opposite direction, i.e. from 

1 Timaeus 37 D, Ε. 2 Timaeus 38 C-E. 

3 Timaeus 36 D, p. 159 above. 4 Aristotle, Metaph. Δ. 12, 1019 a 15. 



1 66 PLATO parti 

east to west. At the same time we are told that the periods in 
which the sun, Venus, and Mercury describe their orbits are the 
same. Thus if, say, Venus and the sun are close together at a 
particular time, they would according to this thepry be nearly 
together again at the end of a year ; but in the meantime Venus, 
moving in a sense contrary to the sun's motion, i.e. in the direction 
of the daily rotation from east to west, would pass through all 
possible angles of divergence from the sun and, after gaining a day, 
would appear with it again. But, as it is, Venus is never far away 
from the sun ; and consequently Plato's statements, thus inter- 
preted, are in evident contradiction to the facts, as easily verified 
by observation. It is not surprising that commentators have 
exhausted their ingenuity to find an interpretation less compromising 
to Plato's reputation as an astronomer. It is true that in the 
Republic all the seven planets revolve in one direction ; but Plato 
is here referring to a phenomenon which is not mentioned in the 
Republic^ namely, the fact that Venus and Mercury respectively 
on the one hand, and the sun on the other, 'overtake and are 
overtaken by one another ', and the idea of the two planets having 
the ' contrary tendency to the sun ' is clearly put forward for the 
precise purpose of explaining this phenomenon. It is accordingly with 
reference to the standings-still and the retrogradations of Venus 
and Mercury that the commentators try to interpret Plato's words. 
On the first passage (36 D) Proclus gives a number of alternatives, 
differing very slightly in substance, some importing the machinery 
of epicycles (which, as Proclus says, are foreign to Plato) and others 
not, but all designed to make Plato refer to nothing more than the 
stationary points and retrogradations; Proclus 1 on this occasion 
rejects them all, observing that the truest explanation is to suppose 
that Plato did not mean that there was any opposition of direction 
among the seven bodies at all, but only that all the seven, moving 
one way, moved in the opposite sense to the general movement of 
the daily rotation. This is cutting the knot with a vengeance. On the 
second passage (38 D) Proclus has the same kind of discussion, 
giving, as an alternative to the importation of epicycles, &c, the 
hypothesis that the ' overtakings ' may be accounted for by the 
speeds of the sun, Venus, and Mercury varying relatively to one 

1 Proclus, In Timaeum 221 D sqq. 



ch. xv PLATO 167 

another at different points of their respective orbits. 1 Chalcidius 2 
has much the same account of the different interpretations, but 
fortunately coupled with a precious passage about the view taken 
by Heraclides of Pontus of the movements of Venus and Mercury 
in relation to the sun : an account which, although it again wrongly 
imports epicycles into Heraclides' theory, as Theon of Smyrna and 
others erroneously import them into Plato's, 3 enables the true theory 
of Heraclides to be disentangled. 4 

Of modern editors Martin 5 refuses to accept any of these explana- 
tions which give a meaning to the passages other than that which 
the words naturally convey, and stoutly maintains that Plato did 
actually say that Venus and Mercury describe their orbits the 
contrary way to the motion of the sun, and meant what he said, 
incomprehensible as this may appear. He quotes in support of his 
view the evidence of Cicero in the fragments of his translation of 
the Timaeus. It is true that Cicero fences with the expression ' the 
contrary tendency', translating it as ' vim quandam contrariam ', 
where ' quandam ' has nothing corresponding to it in the Greek, but 
merely indicates a certain timidity or hesitation which Cicero felt in 
translating δνναμι? by vis; Cicero, perhaps, may have had some 
idea, such as Proclus had, of a capricious force of some kind causing 
the two planets respectively to go faster at one time and slower at 
another. But by his translation of the other passage about the 
seven smaller circles making up the circle of the Other he shows 
that he interpreted Plato as meaning that some of the planets 
describing these circles move in the opposite direction to the others : 
his words are f contrariis inter se cursibus \ 

Archer- Hind 6 maintains that the phrase ' having the contrary 
tendency to it ' does not mean that Venus and Mercury revolve in 
a direction contrary to the sun. He believes that ' Plato meant the 
Sun to share the contrary motion of Venus and Mercury in relation 
to the other planets '. ' It is quite natural,' he says, ' seeing that 
the sun and the orbits of Venus and Mercury are encircled by the 

1 Proclus, In Timaeum 259 A-C. 

2 Chalcidius, Timaeus c. 97, pp. 167-8 ; c. 109, p. 176, Wrobel. 

3 Theon of Smyrna, p. 186. 12-24. 

4 Hultsch, 'Das astronomische System des Herakleides von Pontes', in 
Jahrbuch der classischen Philologie, 1896, pp. 305-16. 

5 Martin, Etudes sur le Timie, ii, pp. 66-75. 

6 Archer-Hind, Timaeus^ pp. 124-5 n • 



1 68 PLATO part ι 

orbit of the earth, while Plato supposed them all to revolve about 
the earth, that he should class them together apart from the four 
whose orbits really do encircle that of the earth : his observations 
would very readily lead him to attributing to these three a motion 
contrary to the rest.' This seems to be a very large assumption ; 
and indeed there is no evidence that Plato made any distinction 
between the groups of planets which we now call inferior and 
superior ; in his system Venus and Mercury were not even inferior 
to the sun, but above it. Besides, although Archer-Hind's view 
would satisfy the first passage about some of the seven moving in 
the contrary direction to the others, it still does not explain the 
second statement that Venus and Mercury have * the contrary ten- 
dency to it ' (the sun). Accordingly he essays a new explanation. 
' What I believe it ' [the contrary tendency] ' to be may be under- 
stood from the accompanying figure which is copied from part of 
a diagram in Arago's Popular Astronomy.' It represents the 
motion of Venus relatively to the earth during one year as observed 
in 1 7 13, and is a sort of epicycloid with a loop. The 'tendency', 
then, is the 'tendency on the part of Venus, as seen from the earth, 
periodically to retrace her steps'. That is, Archer-Hind's explana- 
tion is really an explanation of retrogradations by the equivalent of 
epicycles, and is therefore no better than the anachronistic explana- 
tions by Proclus and others to the same effect. 

I do not think that Schiaparelli 1 is any more successful in his 
explanations. He suggests that the first passage c seems to allude 
to the retrogradations, or perhaps to the opposite positions (with 
reference to the sun) in which Mars, Jupiter, and Saturn on the one 
hand, and Mercury and Venus on the other, carry out their stand- 
ings-still and their retrogradations'. In the second passage he 
translates the words about the ' contrary tendency ' by ' receiving 
a force contrary to it' 2 (the sun), and he implies that this force is 
really in the sun : ' it might be interpreted simply as a power, which 
the sun seems to have, of making these planets go backward, as if it 
attracted them to itself \ 3 This is not less vague than the explana- 
tions of Proclus and others ; it has the disadvantage also that it is 

1 Schiaparelli, I precursor^ p. 16 note. 

2 ' Ricevendo una forza contraria a lui.' 

8 ' Questo tuttavia si potrebbe interpretare semplicemente di una forza che sem- 
bra avere il Sole, di far retrocedere questi pianeti, quasi li attirasse verso di se.' 



ch.xv PLATO 169 

based on a mistranslation of the Greek. The words mean c having ' 
or 'possessed of (είληχότας) the contrary tendency to the sun', which 
clearly shows that the tendency such as it is resides in the planets 
themselves. 

We pass on to the next passage which is relevant to our subject. 

<" But when each of the beings [the planets] which were to join in 
creating time had arrived in its proper orbit, and they had been as 
animate bodies secured with living bonds and had learnt their 
appointed task, then in the motion of the Other/ which was oblique 
and crossed the motion of the Same and was controlled by it, 
one planet described a larger, and another a smaller circle, and 
those which described the smaller circle went round it more swiftly 
and those which described the larger more slowly ; but because 
of the motion of the Same those which went round most swiftly 
appeared to be overtaken by those which went round more slowly, 
though in reality they overtook them. For the motion of the Same, 
which twists all their circles into spirals because they have two sepa- 
rate and simultaneous motions in opposite senses, is the swiftest of all, 
and displays closest to itself that which departs most slowly from it.' 1 

The spirals are easily understood by reference to the figure on 
p. 160. Suppose a planet to be at a certain moment at the point 
F. It is carried by the motion of the Same about the axis GH, 
round the circle FAEB. At the same time it has its own motion 
along the circle FDEC. After 24 hours accordingly it is not at the 
point F on the latter circle, but at a point some way from F on 
the arc FD. Similarly after the next 24 hours, it is at a point 
on FD further from F; and so on. Hence its complete motion is 
not in a circle on the sphere about GH as diameter but in a spiral 
described on it. After the planet has reached the point on the 
zodiac (as D) furthest from the equator it begins to approach the 
equator again, then crosses it, and then gets further away from it on 
the other side, until it reaches the point on the zodiac furthest from 
the equator on that side (as C). Consequently the spiral is included 
between the two small circles of the sphere which have KD y CL as 
diameters. 

The remark about the overtakings of one planet by another is 
also easily explained. Let us consider the matter with reference to 
two of the seven planets in the wider sense, namely the sun and the 
moon. Plato says that the moon, which has the smaller orbit, 

1 Timaeus 38 E-39 B. 



i7o PLATO parti 

moves the faster, that is, the independent movement of the moon 
in its orbit is faster than the independent movement of the sun in 
its orbit, by which he means that the moon describes its orbit in 
the shorter period. Thus the sun describes its orbit in about 365J 
days ; the moon returns to the same position relatively to the fixed 
stars in 27Ί days, a sidereal month, and relatively to the sun in 29Ί 
days, a synodic month. Now, if we consider the whole apparent 
motion of the sun and moon, i.e. including the daily rotation as 
well as the independent motion, the moon appears to go round the 
earth more slowly than the sun. For at new moon it sets soon after 
the sun. The next day it sets later, the day after later still, and so 
on ; it appears therefore to be gradually left behind by the sun, or 
the sun appears to gain on it daily, that is, the moon ' appears to be 
overtaken ' by the sun. On the other hand, if we consider only the 
relative motion of the sun and moon, i.e. if we leave out of account 
the daily rotation as common to both, the moon, describing its 
orbit more quickly than the sun describes its orbit, gains on the 
sun, that is, ' in reality it overtakes ' the sun, as Plato says. 

'And that there might be some clear measure of the relative 
slowness and swiftness with which they moved in their eight revo- 
lutions, God kindled a light in the second orbit from the earth, 
which we now have named the Sun, in order that it might shine 
most brightly through all the heaven, and that living things, so 
many as was meet, should possess number, learning it from the 
revolution of the Same and uniform. Night then and day have 
been created in this manner and for these reasons, making the 
period of the one and most intelligent revolution ; a month has 
passed when the moon, after completing her own orbit, overtakes 
the sun, and a year when the sun has completed its own circle. 

1 But the courses of the others men have not grasped, save a few 
out of many ; and they neither give them names nor investigate the 
measurement of them one against another by means of numbers, in 
fact they can scarcely be said to know that time is represented by 
the wanderings of these, which are incalculable in multitude and 
marvellously intricate. 

1 None the less, however, can we observe that the perfect number 
of time fulfils the perfect year at the moment when the relative 
speeds of all the eight revolutions accomplish their course together 
and reach their starting-point, being measured by the circle of the 
Same and uniformly moving.' l 

1 Timaeus 39 B-D. 



ch.xv PLATO 171 

The ' month ' in the above passage is the synodic month, the 
period in which the moon returns to the same position relatively 
to the sun. ' The courses of the others ' are the periods of the 
planets, which are not called by separate names like 'year' and 
' month ', and which, Plato says, only a very few astronomers had 
attempted to measure one against another. The description of the 
1 wanderings ' of the planets as ' incalculable in multitude and mar- 
vellously intricate ' is an admission in sharp contrast to the assump- 
tion of the spirals regularly described on spheres of which the inde- 
pendent orbits are great circles, and still more so to the assertion in 
the Laws that it is wrong and even impious to speak of the planets 
as ' wandering ' at all, since ' each of them traverses the same path, 
not many paths, but always one circular path ', 1 For the moment 
Plato condescends to use the language of apparent astronomy, the 
astronomy of observation ; and this may remind us that Plato's 
astronomy, even in its latest form as expounded in the Timaeus and 
the Laws, is consciously and intentionally ideal, in accordance with 
his conception of the true astronomy which ' lets the heavens alone '. 

What was the length of Plato's Great Year ? Adam, 2 in his edition 
of the Republic, makes it to be 36,000 years, a figure which he bases 
on his interpretation of the famous passage in the Republic, Book VIII, 
about the Platonic ' perfect number ', which is there called the * period 
for a divine creature', just as, in the passage of the Timaeus, 'the 
perfect number of time fulfils the perfect year '. The perfect num- 
ber of the Republic being, according to both Adam and Hultsch, 
the square of 3,600, or 12,960,000, Adam connects the perfect year 
with the two periods of the myth in the Politicus? during the first 
of which God accompanies and helps to wheel the revolving world, 
while during the second he lets it go. Each of these periods 
contains ' many myriads of revolutions ', the word for revolutions 
being nepioSwv, the same word as is used in the Republic for the 
1 period for a divine creature '. Now in the Politicas πβρίοδοι, 
' periods ' or ' revolutions ', refers to the revolutions of the world on 
its own axis. Hence Adam infers that the perfect or great year 
consists of 12,960,000 daily rotations or 12,960,000 days. Next, he 
cites the Laws, in which Plato divides the year into 360 days (which 

1 Laws vii. 22, 621 B-D, 822 A. 

2 Adam's Republic, vol. ii, pp. 204 sqq. notes, 295-305. 8 Politicus 270 A. 



172 PLATO parti 

is, it is true, an ideal division). 1 Dividing then 12,960,000 by 360, 
we obtain 36,000 years. Adam seeks confirmation of this in the 
fact that we find the period of 36,000 years sometimes actually 
called the ' great Platonic year ' in early astronomical treatises. 
Thus Sacro-Bosco in his Sphaera says that ' the ninth circle in a 
hundred and a few years, according to Ptolemy, completes one 
degree of its own motion and makes a complete revolution in 
36,000 years (which time is commonly called a great year or 
Platonic year) '. Since a text-book of Ptolemaic astronomy makes 
this statement, Adam infers that Ptolemy or some of his prede- 
cessors had understood the Platonic Number, and that we can 
perhaps trace the knowledge of the Number as far back as 
Hipparchus. For Hipparchus discovered the precession of the 
equinoxes and is supposed to have given 36,000 years as the time 
in which the equinoctial points make a complete revolution ; and 
Adam finds it difficult to believe that Hipparchus was uninfluenced 
by Plato's Number. There is, however, the strongest reason for 
doubting this, because Hipparchus's discovery of precession was 
based on something much more scientific than a recollection of the 
Platonic Number, namely actual recorded observations. It is true 
that Ptolemy estimated the movement of precession at 2§° in 
265 years, i.e. about i° in 100 years, or 36" a year, 2 and it is 
commonly supposed that this is precisely Hipparchus's estimate 3 . 
But it is probable that Hipparchus's estimate was much more 
correct. The evidence of Ptolemy 4 shows that Hipparchus 
found the bright star Spica to be, at the time of his observation 
of it, 6° distant from the autumnal equinoctial point, whereas 
he deduced from the observations recorded by Timocharis that 
Timocharis had made the distance 8°. Consequently the motion 
had amounted to 2° in the period between 283 (or 295) and 
129 B.C., a period of 154 (or 166) years ; this gives about 46-8" 
(or 43*4") a year, which is much nearer than 36" to the true 
value of 50-375 7". It is true that, in a quotation which Ptolemy 

1 Laws vi. 756 B-C, 758 B. 

2 Ptolemy, Syntaxis vii. 2, vol. ii, p. 15. 9-17 Heib. Yet Ptolemy, in another 
place (vii. 3, pp. 28-30), infers from two observations made by Timocharis in 
295 and 283 B.C. respectively that the movement amounted to 10' in about 12 
years, which gives 50" a year. 

3 See Tannery, Recherches sur Vhistoire de Vastronomie ancienne, pp. 265 sqq. 

4 Ptolemy, vii. 2, vol. ii, pp. 12, 13, Heib. 



ch.xv PLATO 173 

makes from Hipparchus's treatise on the Length of the Year, 1 
i° in 100 years is the rate mentioned ; but Tannery points out that 
this is not conclusive, because Hipparchus is in the particular 
passage only giving a lower limit, for he says ' at least one-hun- 
dredth of a degree ' and 'in 300 years the movement would have to 
amount to at least 3°'. It would appear therefore that, if the estimate 
of i°in a hundred years was due to Platonic influence at all, it must 
have been Ptolemy who Platonized rather than Hipparchus. And 
it seems clear that the Great Year of 36,000 years, if we assume it 
to be deducible from the passage of Plato, is certainly not ' best 
explained with reference to precession ' as Burnet supposes. 2 Indeed 
the passage in the Timaens is hardly consistent with this, for the 
Great Year is there distinctly said to be the period after which all 
the eight revolutions', i.e. those of the seven 'planets' as well as 
that of the sphere of the fixed stars, come back to the same relative 
positions ; and the only revolution of the sphere of the fixed stars 
that is mentioned is the daily rotation. 

'The visible form of the deities he made mostly of fire, that 
it might be most bright and most fair to behold, and, likening it 
to the All, he fashioned it like a sphere and assigned it to the 
intelligence of the supreme to follow after it ; and he disposed 
it round about throughout all the heaven, to be an adornment 
of it in very truth, broidered over the whole expanse. And he 
bestowed two movements on each, one in the same place and 
uniform, as remaining constant to the same thoughts about the 
same things, the other a movement forward controlled by the 
revolution of the Same and uniform ; but for the other five move- 
ments he made it motionless and at rest, in order that each star 
might attain the highest order of perfection. 

' From this cause then have been created all the stars that 
wander not but remain fixed for ever, living beings, divine, eternal, 
and revolving uniformly and in the same place ; while those which 
have turnings and wander as aforesaid have come into being on 
the principles which we have declared in the foregoing.' 3 

The deities are of course the stars, and ' the intelligence of the 
supreme' which they follow is the revolution of the circle of 
the Same which holds the mastery over all. The two movements 
common to the fixed stars are (1) rotation about their own axes 

1 Ptolemy, I.e., vol. ii, pp. 15, ιξ>, Heib. 

2 Burnet, Early Greek Philosophy, p. 26 note. 3 Timaeas 40 a -b. 



174 PLATO parti 

and (2) their motion as part of the whole heaven in its daily 
rotation, the first being a motion in one and the same place, the 
other a motion 4 forward ', or of translation, in circles parallel to 
the equator, from east to west. The idea that the fixed stars 
rotate about their own axes is attributed by Achilles to the 
Pythagoreans. 1 The ' other five movements ' (in addition to move- 
ment forward) are movements backward, right, left, up and down. 

Rotation about their own axes is only attributed in express terms 
to the fixed stars ; but Proclus is doubtless right in holding that 
Plato intended to convey that the planets also rotate about their 
own axes, the result of which is that, while the fixed stars have 
two motions, the planets have three, rotation about their own axes, 
revolution about the axis of the universe due to their sharing in 
the motion of the Same, and lastly their independent movements 
in their orbits. The ' turnings ' refer to the fact that, like the sun, 
the planets, moving in the circle of the zodiac, go as far from the 
equator as the tropic of Cancer and then turn, first approaching 
the equator and then passing it, until they reach the tropic of 
Capricorn when they again turn back. 

' But the earth our foster-mother, globed round the axis stretched 
from pole to pole through the universe, he made to be guardian 
and creator of night and day, the first and chiefest of the gods that 
have been created within the heaven. 

* But the circlings of these same gods and their comings alongside 
one another, and the manner of the returnings of their orbits upon 
themselves and their approachings, which of the deities meet one 
another in their conjunctions and which are in opposition, in what 
order they pass before one another, and at what times they are 
hidden from us and again reappearing send, to them who cannot 
calculate their movements, terrors, and portents of things to come — 
to declare all this without visible imitations of these same move- 
ments were labour lost.' 2 

It is mainly upon this passage, combined with a passage of 
Aristotle alluding to it, that some writers have based the theory 
that Plato asserted the earth's rotation about its own axis. There 
is now, however, no possibility of doubt that this view is wrong, 
and that Plato made the earth entirely motionless in the centre 
of the universe. This was proved by Boeckh in his elaborate 

1 Achilles, hagoge in Arati phaenomena^ c. 18 {Uranologium, p. 138 c). 

2 Timaeus 40 b-d. 



ch.xv PLATO j 75 

examination of the whole subject * made in reply to a tract by 
Gruppe 2 , and again in a later paper 3 where Boeckh success- 
fully refuted objections taken by Grote to his arguments. The 
cause of the whole trouble is the ambiguity in the meaning of 
the Greek word which is used of the earth ''globed round the axis '. 
It now appears that ίλλομίνην is the correct reading, although there 
is MS. authority for ςίλλομζνην and βίλλομένην ; but all three words 
seem to be no more than variant forms meaning the same thing 
(literally 'rolled'). Boeckh indeed seems to have gone too far in 
saying that ϊίλλομίνην can only mean ' globed round ' in Plato, 
because no actual use of εΐλλβσθαι or eiXeauat in the sense of rotation 
about an axis or revolution in an orbit round a point can be found 
in the Timaeus or elsewhere in the dialogues ; for, as Teichmuller 4 
points out, βί'λλω is related to ίλιξ (a spiral) and έλίττω (Ionic e /λί'σσω), 
to ' roll ' or ' wind ', which latter word is actually used along with the 
word στρέφςσθαι (' to be turned ') in the Theaetetus. 5 But, while ίλλο- 
μζνην does not exclude the idea of motion, it does not necessarily 
include it ; 6 and the real proof that it does not imply rotation here 
(but only being ' rolled round ' in the sense of massed or packed 
round) is not the etymological consideration, but the fact that the idea 
of the earth rotating at all on its axis is quite inconsistent with the 
whole astronomical system described in the Timaeus. An essential 
feature of that system, emphasized over and over again, is the 
motion of the Circle of the Same which carries every other motion 
and all else in the universe round with it ; this is the daily rotation 
which carries round the earth the sphere of the fixed stars, and it is 
this rotation of the fixed stars once completed which makes a day 
and a night ; cf. the passage ' night and day have been created . . . 
and these are the revolution of the one and most intelligent circuit '. 7 

1 Boeckh, Untersuchungen iiber das kosmische System des Platon, 1852. 

2 Gruppe, Die kosmischen Systeme der Griechen, 185 1. 

3 Boeckh, Kleine Schriften, iii, p. 294 sqq. 

4 Teichmuller, Studien zur Geschichte der Begriffe, 1874, pp. 240-2. 

5 Theaetetus 194 B. 

6 Thus in Sophocles, Antigone 340, ίλΧομενων, used of ploughs, means ' going to 
and fro'; but four instances occurring in Apollonius Rhodius tell in favour of 
Boeckh's interpretation of our passage : i. 129 δβσμοί? IWopevov, where (as in 
ii. 1249 also) ίλλόμβϊ/ο? means fast bound' ; i, 329 IWopevois eVi λαίφ^σι, 'with 
sails furled'' ; ii. 27 IWopcvos nep ή/χίλω, ( hemmed in by a crowd.' Simplicius 
(on De caelo, p. 517, 15) cites Ap. Rh. i. 129 and adds that, even if the word is 
spelt dXXopepos, it still means ilpyop.evo$ (' shut in '), as it does once in a play of 
Aeschylus (now lost). 7 Tifnaeus 39 C. 



ij6 PLATO parti 

If the earth rotated about its axis in either direction, it would not be 
the rotation of the sphere of the fixed stars alone which would 
make night and day, but the sum or difference of the two rotations 
according as the earth rotated in the same or the opposite sense to 
the sphere of the fixed stars ; but there is not a word anywhere 
to suggest any cause but the one rotation of the fixed stars in 
24 hours. 

This being so, how did Aristotle come to write { Some say that, 
although the earth lies at the centre, it is yet wound and moves 
about the axis stretched through the universe from pole to pole, as 
is stated in the Timaezts' 1 ? For three MSS. out of Bekker's five 
add the words και κινύσθαι. ' and is moved \ to ϊλλβσθαι, * is wound ', 
whereas the actual passage in the Timaeus has ΙλΧομίνην and 
nothing more. Alexander 2 held that Aristotle must have been 
right in adding the gloss ' and moves ' because he could not have 
been unaware either of the meaning of Ιλλομίνην or of Plato's 
intention. Simplicius 3 is not so sure, but makes the best excuse he 
can. As the word ίλ\ομ£νην might be interpreted by the ordinary 
person as implying rotation, Aristotle would be anxious to take 
account of the full apparent signification as well as the true one, in 
accordance with his habit of minutely criticizing the language of his 
predecessors with all its possible implications ; he might then be 
supposed to say in this passage (which immediately follows his 
reference to those who held that the earth is not in the centre but 
moves round the central fire): ' And, if any one were to suppose 
that Plato affirmed its rotation in the centre through taking ίλλο- 
μένην (being wound) to mean κινουμ,ίνην (being moved), we should 
at once have another class of persons coming under the more 
general category of those who assert that the earth moves ; for the 
hypotheses will be that the earth moves in one of two ways, either 
round the centre or in the centre ; and the person who understands 
Plato's remark in the sense of the latter hypothesis will be proved 
to be in error.' But Simplicius evidently feels that this is not 
a very satisfactory explanation, for he goes on to suggest the alter- 
native that the words καϊ κινύσθαι, ' and moves ', are an interpola- 

1 Aristotle, De caelo ii. 13, 293 030. 

2 Simplicius on De caelo, p. 518. 1-8, 20-21, ed. Heib. 

3 Simplicius, loc. cit., pp. 518. 9-519. 8. 



ch.xv PLATO 177 

tion ; the passage will then, he says, be easy to understand ; the con- 
trast will be a double one, between those who say that the earth (1) is 
not in the centre but (2) moves about the centre, and those who say 
that (1) it is in the centre and (2) is at rest there. It would not be 
unnatural if an unwise annotator had interpolated the words from 
the passage at the beginning of the next chapter (14), where the 
same remark is made without any mention of the Timaeus : ' for, as 
we said before , some make the earth one of the stars, while others 
place it in the centre and say that it is wound and moves (ιλλ^σθαι 
καϊ κινύσθαι,ζζ before) about the axis through the centre joining the 
poles.' * Archer-Hind 2 is disposed to accept the suggestion that the 
words are interpolated from the later into the earlier passage; but the 
suggestion only helps if Aristotle is referring in the later passage to 
some one other than Plato. Archer- Hind, it is true, thinks that 
the added words in the second passage distinguish the theory there 
stated from Plato's ; but I think this is not so. The theory 
alluded to in both passages is, I think, identically the same, as 
indeed we may infer from the words ' as we said before '. Another 
attempted explanation should be mentioned ; it is to the effect that 
the words ' as is stated in the Timaeus ' in the passage of Aristotle 
refer only to the words * about the axis stretched through the uni- 
verse from pole to pole ' and not to the whole phrase ' it is yet wound 
and moves about the axis, &c.\ This explanation was given, as much 
as 600 years ago, by Thomas Aquinas ; 3 in recent years it has been 
independently suggested by Martin 4 and Zeller, 5 and Boeckh has 
an explanation which comes to the same thing. 6 What seems to 
me to be fatal to it is the word ϊλλζσθαι, ' is wound ', immediately 
preceding; this corresponds to Plato's word ιλλομίνηι/, and it is 
impossible, I think, to suppose that ίλλβσθαι does not, as much as 

1 Aristotle, De caelo ii. 14, 296 a 25. 2 Archer-Hind, Timaeus, p. 133 note. 

8 Dreyer {Planetary Systems, p. 78) was apparently the first to point this out. 
The explanation was put forward in Thomas Aquinas's Comment, in libros 
Aristotelis de caelo, lib. ii, lect. xxi (in S. Tnomae Aquinatis Opera omnia, 
iii, p. 205, Rome, 1886) : ' Quod autem addit, quemadmodum in Timaeo 
scriptum est, referendum est non ad id quod dictum est, revolvi et ?noveri, sed 
ad id quod sequitur, quod sit super statutum polum? 

4 Martin in Mem. de VAcad. des Inscriptions et Belles-Lettres, xxx, 1881, 
pp. 77, 78. 

5 Zeller, ' Ueber die richtige Auffassung einiger aristotelischen Citate,' in 
Sitzungsber. der k. Preuss. Akad. der Wis sense haf ten, 1888, p. 1339. 

6 Boeckh, Das kosmische System des Platon, pp. 81-3. 

1410 Ν 



178 PLATO parti 

the words about the axis, refer to the Timaeus. The only possible 
conclusion left is the earlier one of Martin, 1 in which Teichmuller 2 
agrees, namely that Aristotle deliberately misrepresented Plato for 
the purpose of scoring a point. There are many other instances in 
Aristotle of this ' eristic ' and ' sophistical ' criticism, as Teichmuller 
calls it, of Plato's doctrines. 

Other writers seem to have been misled from the first by Aris- 
totle's erroneous description of the theory of the Timaeus. Cicero, 
in speaking of the rotation of the earth about its axis, says : i And 
some think that Plato also affirmed it in the Timaeus but in some- 
what obscure terms.' 3 Plutarch 4 discusses and rejects this inter- 
pretation. Proclus is also perfectly clear that Plato made the earth 
absolutely at rest : 'Let', he says, ' Heraclides of Pontus, who was 
not a disciple of Plato, hold this opinion and make the earth rotate 
round its axis ; but Plato made it unmoved.' 5 Proclus goes on to 
support this by a good argument. If, he says, Plato had not denied 
motion to the earth, he would not have described his c perfect year ' 
with reference to eight motions only ; he would have had to take 
account of the earth's motion also as a ninth. 

The words 'guardian and creator (φύλακα και δημιονργόν) of 
night and day ' have been thought by some to constitute a difficulty 
on the assumption that the earth abides absolutely unmoved in the 
centre. How, it is asked, can a thing which is purely passive be 
said to ' create ' anything ? Martin 6 furnishes the answer to this. 
If the earth were purely passive, it would not be at rest ; it would 
rotate about its axis once in 24 hours, since it would be carried 
round in the daily revolution of the universe. In order to remain 
at rest, as Plato requires, it has to exert a force in the opposite 
direction equal to that exerted by the daily revolution ; it produces 
day and night therefore by the energy of its resistance which keeps 
it at rest, while it is the ' guardian ' by virtue of its immobility. 
A guardian is, as Boeckh says, 7 one who remains on the spot to 
watch and ward ; this is the role of the earth ; if it deserted its post, 

1 Martin, Etudes sur le Timte, ii, p. 87. 

2 Teichmuller, Studien zur Geschichte der Begriffe, 1874, pp. 238-45. 

3 Cicero, Acad. Pr. ii. 39, 123. 4 Plutarch, Quaest. Plat. viii. 1-3, p. 1006 C-F. 
5 Proclus in Tim. p. 281 E. 

β Martin, Etudes sur le Time'e, ii, pp. 88, 90. 

7 Boeckh, Das kosmische System des Platon, p. 69. 



ch.xv PLATO 179 

if it were not there, there would only be light, and not day and 
night ; hence it is called the guardian of night and day. Proclus 
observes that the earth is of course the ' creator ' of night because 
night is the effect of the earth's shadow which is cast in the shape 
of a cone, and the earth can be said to be the creator of day by 
virtue of the day's connexion with night, although one would say 
that the sun rather than the earth is the actual cause of day. 1 
It is, however, the earth which is the cause of the distinction 
between night and day; consequently it may fairly be called the 
1 creator' of both. In the Timaeus Locrus 2 the earth is called 
the 8po? (boundary, limit, or determining principle) of night and day ; 
and Plutarch 3 aptly compares it to the upright needle of the sundial : 
it is its fixedness, he says, which gives the stars a rising and a setting. 
Some expressions in the second paragraph of the passage quoted 
on p. 174 call for a word or two in explanation. The 'circlings', 
&c, are of course those of the planets ; the circlings are their revolu- 
tions round the earth as common centre, as it were in a round dance 
(χορεία), ' their well-ordered and harmonious revolutions,' as Pro- 
clus says. 4 The * comings alongside one another ' (παραβολαί, the 
same word as is used in geometry of the ' application ' of an area to 
a straight line) are explained by Proclus as ' their comings together 
in respect of longitude, while their positions in respect of latitude 
or of depth are different, in other words, their rising simultaneously 
and their setting simultaneously \ e c The returnings of their orbits 
upon themselves and their approachings ' (at των κύκλων προ? εαυ- 
τού? ίπανακνκλήσει? και προσχώρησα?) are somewhat differently 
interpreted. Proclus understands them as meaning retrogradations 
and advance movements respectively : ' for when they advance they 
are approaching their αποκατάσταση (their return to the same place 
in the heavens) ; and, when their movement is retrograde, they return 
upon themselves.' 6 Boeckh agrees in taking προσχωρήσει? to mean 
their return to the same position in the heavens (άποκατάστασι?) 
but takes επανακνκλήσει?, their return upon one another, to be an 
earlier stage of the same motion ; they * turn upon themselves ' in 

1 Proclus in Tim. 282 B, c ; cf. Archer- Hind, Timaeus ; p. 134 note. 

2 Timaeus Locrus 97 D. 

3 Plutarch, Quaest. Plat, viii.3, p. 1006 F ; cf. Defac. in orbe lunae, c. 25, p. 938 E. 

4 Proclus in Tim. p. 284 B. 

6 Ibid., p. 284 C. 6 Ibid. 

Ν 2 



<■■■ 



1 8o PLATO parti 

respect of the circular motion tending to bring them round again to 
the same point, and the ' approaching ' is the arrival at the same 
point. 1 

Some allusions to the sun, moon, and planets as the ' instruments 
of time ' (όργανα χρόνου) bring us to the end of the astronomy of 
the Timaeus. After a passage about the created gods and other 
gods born of them, the Creator makes a second blending of Soul. 

c And when he had compounded the whole, he portioned off souls 
equal in number to the stars and distributed a soul to each star and, 
setting them in the stars as in a chariot, he showed them the nature 
of the universe and declared to them its fated laws .... and how 
they must be sown into the instruments of time befitting them 
severally.' 2 

Archer-Hind explains that the * souls ' here distributed among 
the stars, one to each, are different from the souls of the stars them- 
selves and are rather portions of the whole substance of soul ; this 
was so distributed in order that it might learn the laws of the 
universe; then finally, he thinks, it was redistributed among 
the planets for division into separate souls incorporated in bodies. 3 
The instruments of time are mentioned again a little later on : 

* And when he had ordained all these things for them . . . God 
sowed some in the earth, some in the moon, and some in the other 
instruments of time.' 4 

Gruppe seizes upon this passage to argue that the earth is in- 
cluded with the moon and the other planets among the ' instruments 
of time ', and hence that, as a measurer of time, the earth cannot be 
at rest but must rotate round its axis. But Boeckh 5 points out 
that even in this passage the earth itself need not be an instrument 
of time, for ' the other instruments of time ' may mean ' other than 
the moon* just as well as ' other than the earth and moon '; and it 
is clear from another passage that the earth is not one of the 
1 instruments of time ', For in a sentence already quoted we are 
told that c the sun and the moon and five other stars which have the 
name of planets have been created for defining and . preserving 

1 Boeckh, Das kosmische System des Platon, p. 6o. 

2 Timaeus 41 d, E. 3 Archer- Hind, Timaeus, p. 14 1 -2 note. 
4 Timaeus 42 D. 

6 Boeckh, Das kosmische System des Platon, pp. 71-3. 



ch.xv PLATO 181 

the numbers of time ', 1 i. e. as the instruments of time, It is true 
that the remaining ' instrument ', which measures the day of about 
24 hours, is not here mentioned ; but, when it does come to be 
mentioned, this instrument is not the earth, but the motion of the 
circle of the Same, or the sphere of the fixed stars : ' Night and day 
. . . are one revolution of the undivided and most intelligent circuit.' 2 

We have next to inquire whether still later dialogues contain or 
indicate any modification of the system of the Timaeus. We come 
then to the Laws. 

In Book VII occurs the passage already alluded to above, which 
in the first place exposes what appeared to Plato to be errors in the 
common notions about the movements of the planets current in his 
time, and then states, in a matter of fact way, the view which seems 
to him the most correct. After arithmetic and the science of calcu- 
lation, and geometry as the science of measurement, with the dis- 
tinction between commensurables and incommensurables, astronomy 
is introduced as a subject for the instruction of the young, when the 
following conversation takes place between the Athenian stranger 
and Clinias. 

1 Ath. My good friends, I make bold to say that nowadays we Greeks 
all affirm what is false of the great gods, the sun and the moon. 

CI. What is the falsehood you mean ? 

Ath. We say that they never continue in the same path, and that 
along with them are certain other stars which are in the same case, 
and which we therefore call planets. 

CI. By Zeus, you are right, Ο stranger ; for many times in my 
life I too have noticed that the Morning Star and the Evening 
Star never follow the same course but wander in every possible way, 
and of course the sun and the moon behave in the way which is 
familiar to everybody. 

Ath. These are just the things, Megillus and Clinias, which I say 
citizens of a country like ours and the young should learn with regard 
to the gods in heaven ; they should learn the facts about them all 
so far as to avoid blasphemy in this respect, and to honour them 
at all times, sacrificing to them and addressing to them pious 
prayers. 

CI. You are right, assuming that it is at all possible to learn that 
to which you refer ; if there is anything in our present views about 

1 Timaeus 38 C. 2 Ibid. 39 B, C. 



1 82 PLATO parti 

the gods that is not correct, and instruction will correct it, I too 
agree that we ought to learn a thing of such magnitude and impor- 
tance. Do you then try your best to explain how these things are 
as you say, and we will try to follow your instruction. 

Ath. Well, it is not easy to grasp what I mean, nor yet is it very 
difficult or a very long business. And the proof of this is that, 
although it is not a thing I learnt when I was young or have known 
a long time, I shall not take long to explain it to you ; whereas, if it 
had been difficult, I at my age should never have been able to 
explain it to you at yours. 

CI. I dare say. But what sort of doctrine is this you speak of, 
which you call surprising, and proper to be taught to the young, 
but which we do not know ? Try to tell us this much about it as 
clearly as you can. 

Ath. I will try. Well, my good friends, this view which is held 
about the moon, the sun, and the other stars, to the effect that they 
ever wander, is not correct, but the very contrary is the case. For 
each of them traverses the same path, not many paths \ but always 
one, in a circle, whereas it appears to move in many paths. And 
again, the swiftest of them is incorrectly thought to be the slowest, 
and vice versa. Now, if the truth is one way and we think another 
way, it is as if we had the same idea with regard to horses or long- 
distance runners at Olympia and were to address the swiftest as the 
slowest, and the slowest as the swiftest, and to award the praise 
accordingly, notwithstanding that we knew that the so-called loser 
had really won. I imagine that in that case we should not be 
awarding the praise in the proper way or a way agreeable to the 
runners, who are only human. When then we make this very same 
mistake with regard to the gods, should we not expect that the 
same ridicule and conviction of error would attach to us here and 
in this question as we should have suffered on the racecourse ? 

CI. Nay, it would be no laughing matter at all. 

Ath. No, nor would it be consistent with respect for the gods, if 
we repeated a false report against them.' x 

The sentence italicized above is cited by Gruppe as another 
argument in favour of his hypothesis that Plato attributed to the 
earth rotation about its axis. Plato says that the apparent multi- 
plicity of the courses of each planet is an illusion, and that each has 
one path only. Now, says Gruppe, this is only true if we reject the 
motion of the sphere of the fixed stars as only apparent, and substi- 
tute for it the rotation of the earth round its axis ; for only then 
can it be said, e.g., that the sun and moon have only one movement 

1 Plato, Laws vii. 821 B-822 c. 



ch.xv PLATO 183 

in a circle. If we assume the actual motion of the sun along with 
the sphere of the fixed stars, while the earth remains at rest, the 
circle becomes a spiral as described in the Timaeus. Schiaparelli, 1 
influenced also by the passage of Aristotle which he thinks repre- 
sents what Aristotle must have known to be the final view of Plato 
through hearing the matter discussed in his school, accepts Gruppe's 
conclusion, not apparently having been aware, at the time that he 
did so, of Boeckh's complete refutation of it. Boeckh 2 answers in 
the first place that the unity of the movement of the planets 
in single circles is not supposed, here any more than in the Timaeus •, 
to be upset by the fact that the movement of the circle of the Same 
turns them into spirals. Thus in the Timaeus^ in the very next 
sentence but one to that about the spirals, Plato speaks of the moon 
as describing ' its own circle ' in a month, and of the sun as describing 
* its own circle ' in a year. Similarly, Dercyllides 3 says that the 
orbits of the planets are primarily simple and uniform circles 
round the earth ; the turning of these circles into spirals is merely 
incidental. 

Gruppe goes so far as to find the heliocentric system in the 
passage before us, by means of a forced interpretation of the words 
about the planets which are really the quickest being regarded as 
the slowest and vice versa. He relies in the first place on two 
passages of Plutarch as follows: (1) ' Theophrastus also adds that 
Plato in his old age regretted that he had given the earth the 
middle place in the universe, which was not appropriate to it/ 4 and 
(2) 'they say that Plato in his old age was moved by these con- 
siderations [the Pythagorean theory of the central fire] to regard 
the earth as placed elsewhere than in the centre, and the middle 
and chiefest place as belonging to some worthier body ' ; 5 he then 
straightway proceeds to assume the worthier body to be the sun, 
and the ambiguity as regards swiftness and slowness to refer to the 
stationary points and the retrogradations of the planets. Schia- 
parelli, 6 however, points out, as Boeckh 7 had done before him, that 

1 Schiaparelli, I precursori^ p. 19 sq. 

2 Boeckh, Das kosmische System, des Platon, pp. 52 sqq. 

3 Theon of Smyrna, p. 200. 23 sq. 

4 Plutarch, Quaest. Plat. viii. 1, p. 1006 C. 

5 Plutarch, Numa, c. 11. 6 Schiaparelli, I precursorz, p. 21. 
7 Boeckh, Das kosmische System des Platon, p. 57. 



1 84 PLATO parti 

this cannot be correct, as it is indicated a little earlier in the same 
passage of the Laws that everybody sees the same phenomena 
illustrated in the case of the sun and moon : this clearly implies, 
first, that the sun moves and, secondly, that the irregularities 
cannot be retrogradations, seeing that they do not exist in the case 
of the sun and moon. The fact is that the ambiguity pointed out 
in the Laws with regard to the speed of the planets is exactly the 
same as that which we have read of in the Timaeus, and that 
the passage in the Laws changes nothing whatever in the system 
expounded in the earlier dialogue. The remarks quoted from 
Plutarch will be dealt with later. 

But we have not even yet finished with the arguments as regards 
the supposed rotation of the earth in Plato's final system. Schia- 
parelli 2 finds another argument in its favour in the Epinomis, a. 
continuation of the Laws attributed to Philippus of Opus, a disciple 
of Plato, who is also said to have revised and published the Laws, 
which had been left unfinished. The system described in the 
Epinomis is the same as the system of the Timaeus. There are 
eight revolutions. Two are those of the moon and the sun ; 3 then 
come two others, those of Venus and Mercury, of which it is said 
that their periods are about the same as that of the sun, 4 so that 
no one of the three can be said to be slower or faster than the 
others ; 5 after these are mentioned the three revolutions of the other 
planets, Mars, Jupiter, and Saturn, which are said to travel in the 
same direction as the sun and moon, i.e. from west to east. The 
eighth revolution is not that of the earth, so that here, as in the 
Timaetis and the Laws, no rotation is attributed to the earth. Of 
the eighth revolution we read : 

1 And one of the moving bodies, the eighth, is that which it is most 
usual to call the universe above [i.e. the sphere of the fixed stars], 
which travels in the opposite sense to all the others, while carrying 
the others with it, as men with little knowledge of these things would 
suppose. But whatever we adequately know we must affirm and we 
do affirm.' 6 

Upon this Schiaparelli remarks, adverting to the italicized words : 
1 Plato then declares, in the Epinomis also, that men who under- 

1 Laws 821 C. 2 Schiaparelli, I precursory pp. 20-21. 

3 Epinomis 986 A, B. 4 Ibid. 986 E, 987 Β. β Ibid. 987 Β. 6 Ibid. 987 B. 



ch. xv PLATO 185 

stand little about astronomy believe in the daily revolution of the 
heaven. If he expresses himself according to this system, it is for 
the purpose of adapting himself to the intelligence of the ordinary 
person. Here we have what Aristotle ^doubtless had in mind when 
he wrote his celebrated remark about the rotation of the earth.' 1 
But if this is the meaning of the passage, why did the author, after 
saying (apparently by way of contrast) ' but what we adequately 
know we must affirm and do affirm ',stop there and say not a single 
word of any alternative to the general rotation of the * heaven ' ? 
There is still not a word of the earth's rotation, and indeed it is 
excluded by the limitation of the revolutions to eight, as remarked 
above. We must therefore, I think, reject Schiaparelli's interpreta- 
tion of the passage and seek another. It occurs to me that the 
emphasis is on the word ' men * (άνθρωποι.? without the article), and 
that the meaning is ' so far as mere human beings can judge, who 
can have little knowledge of these things '. The words immediately 
following are then readily intelligible ; they would mean ' but if we 
are reasonably satisfied of a thing we must have the courage to state 
our view \ 2 

One other passage of the Epinomis is quoted by Martin 3 as 
evidence that it only repeats the theory of the Timaeus without 
change. All the stars are divine beings with body and soul. A 
proof that stars have intelligence is furnished by the fact that * they 
always do the same things, because they have long been doing 
things which had been deliberated upon for a prodigious length of 
time, and they do not change their plans up and down, do one 
thing at one time and another at another, or wander and change 
their orbits'. 4 Consequently, as the stars include the planets, the 
Epinomis, like the Timaeus? seems to deny the distinction between 
perigee and apogee, all variations of angular speed, stationary posi- 
tions and retrogradations, and all movement in celestial latitude. 

We have, lastly, to consider the two passages of Plutarch quoted 
above (p. 183) to the effect that Plato is said to have repented in his 

1 Schiaparelli, 1 precursor i, pp. 20-1. 

2 Cf. Laws 716 c, to the effect that God is the real measure of all things, 
much more so than any man. 

3 Martin in Mem. de I 1 Acad. des Inscriptions et Belles- Lettres> xxx, 188 1, p. 90. 

4 Epinomis 982 c, D. 

5 Timaeus 40 Β ; cf. 34 A, 43 B, &c. 



1 86 PLATO parti 

old age of having put the earth in the centre instead of assigning 
the worthier place to a worthier occupant. These passages have 
been fully dealt with by Boeckh 1 and by Martin 2 after him, and it 
is difficult or impossible to dissent from their conclusion, which is 
that the tradition is due to a misunderstanding and is unworthy 
of credence. To begin with, although the Laws is later than the 
Timaeus and so late that Plato did not finish it, there is in it no sign 
of a change of view. Nor is there any sign of such in the Epinomis 
written by Plato's disciple, Philippus of Opus ; and it is incredible 
that, if the supposed change of view had come out in the last oral 
discussions with the Master, Philippus would not have known about 
it and mentioned it. Even assuming the tradition to be true, we 
can at all events reject without hesitation the inference of Gruppe 
that the sun was Plato's new centre of the universe. If the sun had 
been the centre, this would surely have been stated, and we should 
not have been put off with the vague phrase ' some worthier occu- 
pant '. As, in the Numa where this expression occurs, Plutarch has 
just been speaking of the central fire of the Pythagoreans, the 
natural inference is that Plato's new centre, if he came to assume 
one at all, would be either the Pythagorean central fire or some 
imaginary centre of the same sort. But from what source did 
Theophrastus get the story which he repeats? Obviously from 
hearsay, since there is not a particle of written evidence to confirm 
it. The true explanation seems to be that some of Plato's imme- 
diate followers in the Academy altered Plato's system in a 
Pythagorean sense, and that the views of these Pythagorizing 
Platonists were then put down to Plato himself. In confirmation 
of this Boeckh quotes the passage of Aristotle in which, after 
speaking of the central fire of the Pythagoreans and the way in 
which they invented the counter-earth in order to force the pheno- 
mena into agreement with their preconceived theory, he goes on to 
indicate that there was in his time a school of philosophers other 
than the Pythagoreans who held a similar view : ' And no doubt 
many others too would agree (with the Pythagoreans) that the place 
in the centre should not be assigned to the earth, if they looked for 

1 Boeckh, Das kosmische System des P/aton, pp. 144-50. 

2 Martin in Mem. de VAcad. des Inscriptions et Be/tes-Lettres, xxx, 1881, 
pp. 128-32. 



ch.xv PLATO 187 

the truth, not in the observed facts, but in a priori arguments. For 
they hold that it is appropriate to the worthiest object that it 
should be given the worthiest place. Now fire is worthier than earth, 
the limit worthier than the things which are between the limits, 
while both the extremity and the centre are limits : consequently, 
reasoning from these premises, they hold that it is not the earth 
which is placed at the centre of the sphere, but rather fire.' 1 
Simplicius 2 observes upon this that Archedemus, who was younger 
than Aristotle, held this view, but that, as Alexander says, it is 
necessary to inquire historically who were the persons earlier than 
Aristotle who also held it. As Alexander could not find any such, 
he concluded that it was not necessary to suppose that there were any 
except the Pythagoreans. But the present indicative ' they hold ' 
makes it clear that Aristotle had certain other persons in mind who, 
however, were not philosophers of an earlier time but were contem- 
poraries of his own. These may well have been members of, or an 
offshoot from, the Academy who expressed the views in question, 
not in written works, but in discussion ; and, if this were so, nothing 
would be more natural than that a tradition which referred to the 
views of these persons should be supposed to represent the views of 
Plato in his old age. 

Tannery 3 has a different and very ingenious explanation of 
Theophrastus's dictum about Plato's supposed change of view. 
This explanation is connected with Tannery's explanation of 
another mystery, that of the attribution to one Hicetas of Syracuse 
of certain original discoveries in astronomy. Diogenes Laertius 4 
says of Philolaus that ' he was the first to assert that the earth 
moves in a circle, though other authorities say that it was Hicetas 
the Syracusan \ Aetius says 5 that ' Thales and those who followed 
him said that the earth was one; Hicetas the Pythagorean that 
there were two, our earth and the counter-earth \ From these two 
passages taken together we should naturally infer that Hicetas was 
by some considered to be the real author of the doctrine attributed 

1 Aristotle, De caelo ii. 13, 293 a27~b 1. 

2 Simplicius on De eaelo, p. 513, ed. Heib. 

s Tannery, 'Pseudonymes antiques' in Revue des Etudes grecques, x, 1897, 
pp. 127-37. 

4 Diog. L. viii. 85 (Vors. i 2 , p. 233. 33). 

5 Aet. iii. 9. 1. 2 (D. G. p. 376 ; Vors. i 2 , p. 265. 25). 



1 88 PLATO parti 

to Philolaus, in which the earth and counter-earth, along with the 
sun, moon, and planets, revolve round the central fire. Cicero, 1 
however, has a different story : ' Hicetas of Syracuse, as Theo- 
phrastus says, holds that the heaven, the sun, the moon, the stars, 
and in fact all things in the sky remain still, and nothing else in 
the universe moves except the earth ; but, as the earth turns and 
twists about its axis with extreme swiftness, all the same results 
follow as if the earth were still and the heaven moved.' This is 
of course not well expressed, because, on the assumption that the 
earth rotates about its axis once in every 2,4 hours, the sun, moon, 
and planets would not in fact remain at rest any more than on the 
assumption of a stationary earth, for they would still have their 
independent movements ; but Cicero means no more than that the 
rotation of the earth is a complete substitute for the apparent daily 
rotation of the heaven as a whole. However, the passage clearly 
implies that Hicetas asserted the axial rotation of the earth, and 
not its revolution with the counter-earth, &c, round the central 
fire. The statements therefore of Cicero on the one side, and of 
Diogenes and Aetius on the other, are inconsistent. Tannery 
agrees with Martin 2 that we must accept as the more correct the 
version of Diogenes and Aetius identifying Hicetas with the theory 
commonly attributed to Philolaus. Now, says Tannery, Aristotle, 
when speaking of the doctrine of the central fire as that of * the 
philosophers of Italy, the so-called Pythagoreans', clearly shows 
that he did not attribute the doctrine to the Pythagoreans in 
general or to Philolaus ; if he had seen the book of Philolaus of 
which our fragments formed part, and if he had referred to that 
work in this passage, he would have spoken of Philolaus by name 
instead of using the circumlocution; hence Aristotle must have 
been quoting from a book by some contemporary purporting to 
give an account of Pythagorean doctrines or doctrines claiming 
to be such. Tannery supposes therefore that Aristotle was referring 
to Hicetas, and that Hicetas was one of the personages in a certain 
dialogue, in which Hicetas represented the system known by the 
name of Philolaus, while Plato was his interlocutor. This dialogue 

1 Cicero, Acad. Pr. ii. 39. 123 (Vors. i 2 , p. 265. 20). 

2 Martin, Etudes sur le Timee, vol. ii, pp. 101, 125 sq. 



CH. XV PLATO 189 

would be one of those written by Heraclides of Pontus. One of 
Heraclides' dialogues was * On the Pythagoreans ', and an account 
of the system of the central fire might easily form part of one of the 
others, e.g. that 'On Nature' or On the things in heaven'. Now 
there was a historical personage of the name of Hicetas of Syracuse 
whom Plato might well have known. He was a friend of Dion and 
he appears in Plutarch's lives of Dion and Timoleon as a political 
personage of some importance. Faithful to Dion, and for a time to 
his family after Dion's assassination, he threw over that family and 
seized the tyranny at Leontini, remaining the principal adversary of 
Dionysius the Younger until the arrival of Timoleon, when he was 
conquered and killed by the latter. There is nothing to suggest 
that he was a physicist or a Pythagorean ; but he might quite well 
be represented in the dialogue as one who knew by oral tradition 
the doctrines of the school, and was therefore a suitable interlocutor 
with Plato. Plato's remarks in the dialogue might no doubt easily 
indicate a change from the views which we find in his own dialogues, 
and this is a possible explanation of the misconception on the part 
of Theophrastus and the Doxographi. Tannery adds that, on his 
hypothesis, we can hardly any longer consider the so-called Philolaic 
system as anything else but a brilliant phantasy due to that clever 
raconteur Heraclides. I do not see the necessity for this, and it is 
extremely difficult to believe that Heraclides invented both the 
theory of Philolaus and his own theory of the rotation of the earth 
about its axis ; I do not see why we should not suppose that the 
system known by the name of Philolaus actually belonged to him 
or to the Pythagoreans proper, and that Hicetas represented the 
Pythagorean view rather than a new discovery of Heraclides. 
Tannery's attractive hypothesis is accepted by Otto Voss. 1 

1 Otto Voss, De Heraclidis Pontici vita et scrip tis, Rostock, 1896, p. 64. 



XVI 

THE THEORY OF CONCENTRIC SPHERES. 
EUDOXUS, CALLIPPUS, AND ARISTOTLE 

Diogenes Laertius 1 tells us that Eudoxus of Cnidus was 
celebrated as geometer, astronomer, physician, and legislator. 
Philosopher and geographer in addition, he commanded and en- 
riched almost the whole field of learning ; no wonder that (though 
it was a poor play on his name) he was called ei /δοξο? (' celebrated ') 
instead of Eudoxus. In geometry he was a pupil of Archytas of 
Tarentum, and it is clear that he could have had no better instructor, 
for Archytas was a geometer of remarkable ability, as is shown by 
his solution of the problem of the two mean proportionals handed 
down by Eutocius. 2 This solution furnishes striking evidence of 
the boldness and breadth of conception which already characterized 
Greek geometry, seeing that even in that early time it did not 
shrink from the use of complicated curves in space produced by the 
intersection of two or more solid figures. Archytas solved the pro- 
blem of the two mean proportionals by finding a point in space as 
the intersection of three solid figures. The first was an anchor-ring 
or tore with centre C, say, inner radius equal to zero, and outward 
radius 2 a, say ; the second was a right cylinder of radius a so placed 
that its surface passes through the centre C of the tore and its axis 
is parallel to the axis of the tore or perpendicular to the plane 
bisecting the tore in the same way as a split ring is split ; the third 
surface was a certain right cone with C as vertex. The intersection 
of the first two surfaces gives of course a curve (or curves) of double 
curvature in space, and the third surface cuts it in points, one of 
which gives Archytas what he seeks. There is, as we shall see, 

1 Diog. L. viii. 86-91. 

2 See Heiberg's Archimedes, vol. iii, pp. 98-102; or my Apollonius of Perga, 
pp. xxii, xxiii. 



EUDOXUS 191 

a remarkable similarity between this construction and the way in 
which Eudoxus's ' spherical lemniscate ' (hippopede) is evolved as the 
intersection between a sphere, a cylinder touching it internally, and 
a certain cone, so that we may well believe that Eudoxus owed 
much to Archytas. To Eudoxus himself geometry owes a debt 
which is simply incalculable, and it is doubtful, I think, whether, for 
originality and power, any of the ancient mathematicians except 
Archimedes can be put on the same plane with him. Although no 
geometrical work of Eudoxus is preserved, there is, in the first 
place, a monument to him aere perennius in Book V of Euclid's 
Elements ; it was Eudoxus who invented and elaborated the great 
theory of proportion there set out, the essence of which is its 
applicability to incommensurable as well as commensurable quan- 
tities. The significance of this theory of proportion, discovered 
when it was, cannot be over-rated, for it saved geometry from the 
impasse into which it had got through the discovery of the irrational 
at a time when the only theory of proportion available for geo- 
metrical demonstrations was the old Pythagorean numerical theory, 
which only applied to commensurable magnitudes. Nor can any 
one nowadays even attempt to belittle the conception of equal 
ratios embodied in Euclid V, Def. 5, when it is remembered that 
Weierstrass's definition of equal numbers is word for word the same, 
and Dedekind's theory of irrational numbers corresponds exactly 
to, nay, is almost coincident with, the same definition. Eudoxus's 
second great discovery was that of the powerful method of 
exhaustion which not only enabled the areas of circles and the 
volumes of pyramids, cones, spheres, &c, to be obtained, but is at 
the root of all Archimedes' further developments in the mensuration 
of plane and solid figures. It is not then surprising that Eudoxus 
should have invented a geometrical hypothesis for explaining the 
movements of the planets which for ingenuity and elegance yields 
to none. 

Eudoxus flourished, according to Apollodorus, in 01. 103 = 368- 
365 B.C., from which we infer that he was born about 408 B.C., and 
(since he lived 53 years) died about 355 B.C. In his 23rd year he 
went to Athens with the physician Theomedon, and there for two 
months he attended lectures on philosophy and oratory, and in 
particular the lectures of Plato ; so poor was he that he took up his 



igt THEORY OF CONCENTRIC SPHERES parti 

abode at the Piraeus and trudged to Athens and back on foot each 
day. It would appear that his journey to Italy and Sicily to study 
geometry with Archytas and medicine with Philistion must have 
been earlier than the first visit to Athens at 23, for from Athens he 
returned to Cnidus, after which he went to Egypt with a letter of 
introduction to the king Nectanebus, given him by Agesilaus ; the 
date of this journey was probably 381-380, or a little later, and he 
stayed in Egypt sixteen months. After that he went to Cyzicus, 
where he collected round him a large school with whom he migrated 
to Athens in 368 B.C. or a little later. There is apparently no 
foundation for the story mentioned by Diogenes Laertius that he 
took up a hostile attitude to Plato, nor, on the other side, for the 
stories that he went with Plato to Egypt and spent thirteen years in 
the company of the Egyptian priests, or that he visited Plato when 
Plato was with Dionysius, i. e. the younger Dionysius, on his third 
visit to Sicily in 361B.C. Returning later to his native place, 
Eudoxus was by a popular vote entrusted with legislative office. 

When in Egypt Eudoxus assimilated the astronomical knowledge 
of the priests of Heliopolis and himself made observations. The 
observatory between Heliopolis and Cercesura used by him was 
still pointed out in Augustus's time; 1 he also had one built at 
Cnidus, from which he observed the star Canopus which was not 
then visible in higher latitudes. 2 He wrote two books entitled 
respectively the Mirror (ίνοπτρον) and the Phaenomena : the poem 
of Aratus was, so far as verses 19-732 are concerned, drawn from 
the Phaenomena of Eudoxus. It is probable that he also wrote 
a book on Sphaeric, dealing with the same subjects as Autolycus's 
On the moving sphere and Theodosius's Sphaerica. 

In order to fix approximately the positions of the stars, including 
in that term the fixed stars, the planets, the sun, and the moon, 
Eudoxus probably used a dioptra of some kind, though doubtless of 
more elementary construction than that used later by Hipparchus ; 

1 Strabo, xvii. 1. 30, pp. 806-7 Cas. 

2 Strabo, ii. 5. 14, p. 119 Cas. Hipparchus {In Arati et Eudoxi phaeno7nena 
Commentariorum libri ires, p. 114, 20-28) observes that Eudoxus placed the 
star Canopus exactly on the 'always invisible circle', but that this is not correct, 
since at Rhodes the circumference of this circle is 36 and at Athens 37 from 
the South pole, while Canopus is about 38^° distant from that pole, so that 
Canopus is seen in Greece north of the said circle. But, at the time when this 
was written, Hipparchus had not yet discovered Precession. 



ch.xvi EUDOXUS 193 

and he is credited with the invention of the arachne (spider's web), 
which, however, is alternatively attributed to Apollonius, 1 and which 
seems to have been a sun-clock of some kind. 2 

But it was on the theoretic even more than the observational side 
of astronomy that Eudoxus distinguished himself, and his theory 
of concentric spheres, by the combined movements of which he 
explained the motions of the planets (thereby giving his solution 
of the problem of accounting for those motions by the simplest of 
regular movements), may be said to be the beginning of scientific 
astronomy. 

Two pupils of Eudoxus achieved fame, one in geometry, 
Menaechmus, the reputed discoverer of the conic sections, and the 
other in astronomy, Helicon of Cyzicus, who was said to have 
successfully predicted a solar eclipse. 

The ancient evidence of the details of Eudoxus's system of con- 
centric spheres (which he set out in a book entitled On speeds, Ilepl 
ταγών, now lost) is contained in two passages. The first is in 
Aristotle's Metaphysics? where a short notice is given of the num- 
bers and relative positions of the spheres postulated by Eudoxus 
for the sun, moon, and planets respectively, the additions which 
Callippus thought it necessary to make to the numbers of the spheres 
assumed by Eudoxus, and lastly the modification of the system 
which Aristotle himself considers necessary ' if the phenomena are 
to be produced by all the spheres acting in combination '. A more 
elaborate and detailed account of the system is contained in Sim- 
plicius's commentary on Book II of the De caelo of Aristotle ; 4 
Simplicius quotes largely from Sosigenes the Peripatetic (second 
century A.D., the teacher of Alexander Aphrodisiensis, not the 
astronomer who assisted Caesar in his reform of the calendar), 
observing that Sosigenes drew from Eudemus, who dealt with the 
subject in the second book of his History of Astronomy. b Ideler 
was the first to appreciate the elegance of the theory and to attempt 



1 Vitruvius, De architect, ix. 8 (9). I. 

2 Bil finger, Die Zeitmesser der antiken V'olker, p. 22. 

3 Aristotle, Metaph. Λ. 8, 1073 b 17 - 1074 a 14. 

4 Simplicii in Aristotelis de caelo co?nmentaria, p. 488. 1 8-24, pp.493. 4 - 5c6. 
18, Heiberg; p. 498 a 45-b 3, pp. 498 b 27-503 a 33, Brandis. 

6 Simpl. on De caelo, p. 486, 18-21, Heib. ; p. 498 a 46-8, Brandis. 

1410 Ο 



194 THEORY OF CONCENTRIC SPHERES parti 

to explain its working; 1 he managed by means of an ordinary- 
globe to indicate roughly how Eudoxus explained the stationary 
points and retrogradations of the planets and their movement in 
latitude. E. F. Apelt 2 too gave a fairly full exposition of the 
theory in a paper of 1849. But it was reserved for Schiaparelli to 
work out a complete restoration of the theory and to investigate in 
detail the extent to which it could account for the phenomena ; this 
Schiaparelli did in a paper which has become classical, 3 and which 
will no doubt be accepted by all future historians (in the absence of 
the discovery of fresh original documents) as the authoritative and 
final exposition of the system. 4 

The passages of Aristotle and Simplicius are translated in full 
in Appendices I and II to Schiaparelli's paper. The former may 
properly be reproduced here. 

* Eudoxus assumed that the sun and moon are moved by three 
spheres in each case ; the first of these is that of the fixed stars, the 
second moves about the circle which passes through the middle of 
the signs of the zodiac, while the third moves about a circle 
latitudinally inclined to the zodiac circle ; and, of the oblique 
circles, that in which the moon moves has a greater latitudinal 
inclination than that in which the sun moves. The planets are 
moved by four spheres in each case ; the first and second of these 
are the same as for the sun and moon (the first being the sphere of 
the fixed stars which carries all the spheres with it, and the second, 
next in order to it, being the sphere about the circle through 
the middle of the signs of the zodiac which is common to all the 
planets 5 ); the third is in all cases a sphere with its poles on 
the circle through the middle of the signs ; the fourth moves about 

1 Ideler, ' Ueber Eudoxus* in Abh.der Berliner Akademie 1 hist.-phil. Classe, 
1828, pp. 189-212, and 1830, pp. 49-88. 

3 E. F. Apelt, 'Die Spharentheorie des Eudoxus und Aristoteles' in the 
Abhandlungen der Fries' schenSchule, Heft ii (Leipzig, 1849). 

3 Schiaparelli, 'Le sfere omocentriche di Eudosso, di Callippo e di Aristotele', 
in Pubblicazioni del R. Osservatorio di Brera in Milano, No. ix, Milano, 1875 ; 
German translation by W. Horn, in Abh. zur Gesch. der Math., I. Heft, Leipzig, 
1877, pp. 101-98. 

* It is true that Martin {Mdm. de Γ Acad, des Inscr. xxx, 1881) took objection 
to Schiaparelli's interpretation of the theories of the sun and moon, but he was 
sufficiently answered by Tannery (' Seconde note sur le systeme astronomique 
d'Eudoxe ' in Mem. de la Soc. des set. fihys. et nat. de Bordeaux, 2 e serie, 
v, 1883, pp. 129 sqq., republished in Paul Tannery ', Memoires scientifiques, ed. 
Heiberg and Zeuthen, vol. i, 1912, pp. 317-38. 

5 άπαπ&ν, with which we must, strictly speaking, understand σφαιρών (spheres) 
or possibly φορών (motions). 



ch.xvi EUDOXUS 195 

a circle inclined to the middle circle (the equator) of the third sphere ; 
the poles of the third sphere are different for all the planets except 
Aphrodite and Hermes, but for these two the poles are the same. 1 

Fuller details are given by Simplicius, but, before we pass to the 
details, we may, following Schiaparelli, here as throughout, inter- 
pose a few general observations on the essential characteristics of 
the system. 2 Eudoxus adopted the view which prevailed from the 
earliest times to the time of Kepler, that circular motion was suffi- 
cient to account for the movements of all the heavenly bodies. 
With Eudoxus this circular motion took the form of the revolution 
of different spheres, each of which moves about a diameter as axis. 
All the spheres were concentric, the common centre being the 
centre of the earth ; hence the name of ' homocentric spheres ' used 
in later times to describe the system. The spheres were of different 
sizes, one inside the other. Each planet was fixed at a point in the 
equator of the sphere which carried it, the sphere revolving at 
uniform speed about the diameter joining the corresponding poles ; 
that is, the planet revolved uniformly in a great circle of the sphere 
perpendicular to the axis of rotation. But one such circular motion 
was not enough ; in order to explain the changes in the speed of 
the planets' motion, their stations and retrogradations, as well as 
their deviations in latitude, Eudoxus had to assume a number of 
such circular motions working on each planet and producing by 
their combination that single apparently irregular motion which 
can be deduced from mere observation. He accordingly held that 
the poles of the sphere which carries the planet are not fixed, 
but themselves move on a greater sphere concentric with the 
carrying sphere and moving about two different poles with a 
speed of its own. As even this was not sufficient to explain the 
phenomena, Eudoxus placed the poles of the second sphere on 
a third, which again was concentric with and larger than the first and 
second and moved about separate poles of its own, and with a speed 
peculiar to itself. For the planets yet a fourth sphere was required 

1 Aristotle, Metaph. Λ. 8, 1073 b 17-32. 

2 A very useful summary of the results of Schiaparelli's paper is given in 
Dreyer's History of the Planetary Systems front Thales to Kepler (Camb. Univ. 
Press, 1906), pp. 90-103. My account must necessarily take the same line ; and 
my apology for inserting it instead of merely referring to Dreyer's chapter on the 
subject must be that a sketch of the history of Greek astronomy such as the 
present would be incomplete without it. 

Ο 2 



196 THEORY OF CONCENTRIC SPHERES parti 

similarly related to the three others ; for the sun and moon he 
found that, by a suitable choice of the positions of the poles and of 
speeds of rotation, he could make three spheres suffice. In the 
accounts of Aristotle and Simplicius the spheres are described in 
the reverse order, the sphere carrying the planet being the last. 
The spheres which move each planet Eudoxus made quite separate 
from those which move the others. One sphere sufficed of course 
to produce the daily rotation of the heavens. Thus, with three 
spheres for the sun, three for the moon, four for each of the planets 
and one for the daily rotation, there were 27 spheres in all. It does 
not appear that Eudoxus speculated upon the causes of these 
rotational motions or the way in which they were transmitted from 
one sphere to another ; nor did he inquire about the material of 
which they were made, their sizes and mutual distances. In the 
matter of distances the only indication of his views is contained in 
Archimedes' remark that he supposed the diameter of the sun to be 
nine times that of the moon, 1 from which we may no doubt infer that 
he made their distances from the earth to be in the same ratio 9:1. 
It would appear that he did not give his spheres any substance or 
mechanical connexion ; the whole system was a purely geometrical 
hypothesis, or a set of theoretical constructions calculated to repre- 
sent the apparent paths of the planets and enable them to be com- 
puted. We pass to the details of the system. 

The moon has a motion produced by three spheres ; the first and 
outermost moves in the same sense as the fixed stars from east to 
west in twenty-four hours ; the second moves about an axis per- 
pendicular to the plane of the zodiac circle or the ecliptic, and in 
a sense opposite to that of the daily rotation, i. e. from west to east ; 
the third moves about an axis inclined to the axis of the second, at 
an angle equal to the highest latitude attained by the moon, and in 
the sense of the daily rotation from east to west ; the moon is fixed 
on the equator of this third sphere. Simplicius observes that the 
third sphere is necessary because it is found that the moon does not 
always reach its highest north and south latitude at the same points 
of the zodiac, but at points which travel round the zodiac in the 
inverse order of the signs. 2 He says at the same time that 

1 Archimedes, ed. Heib., vol. ii, p. 248. 4-8 ; The Works of Archimedes > p. 223. 
8 Simplicius on De caelo, p. 495. 10-13, Heib. 



ch.xvi EUDOXUS 197 

the motion of the third sphere is slow, the motion of the node being 
'quite small during each month ', while he implies that the monthly- 
motion round the heavens is produced by the second sphere, the 
equator of which is in the plane of the zodiac or ecliptic. The 
object of the third sphere was then to account for the retrograde 
motion of the nodes in about 1 8-| years. But it is clear (as Ideler saw) 
that Simplicius's statement about the speeds of the third and second 
spheres is incorrect. If it had been the third sphere which moved 
very slowly, as he says, the moon would only have passed through 
each node once in the course of 223 lunations, and would have been 
found for nine years north, and then for nine years south, of the 
ecliptic. In order that the moon may pass through the nodes as 
often as it is observed to do, it is necessary to interchange the 
speeds of the second and third spheres as given by Simplicius ; that 
is, we must assume that the third sphere produces the monthly 
revolution of the moon from west to east in 27 days 5I1. 5m. 36sec. 
(the draconitic or nodal month) round a circle inclined to the 
ecliptic at an angle equal to the greatest latitude of the moon, 
and then that this oblique circle is carried round by the second 
sphere in a retrograde sense along the ecliptic in a period of 223 
lunations ; lastly, we must assume that both the inner spheres, the 
second and third, are bodily carried round by the first sphere in 24 
hours in the sense of the daily rotation. There can be no doubt 
that this was Eudoxus's conception of the matter. The mistake 
made by Simplicius seems to go back as far as Aristotle himself, 
since, in the passage of the Metaphysics quoted above, Aristotle 
clearly implies that the second sphere corresponds to the move- 
ment in longitude for all the seven bodies including the sun and 
moon, whereas in fact it only does so in the case of the five planets ; 
and no doubt Sosigenes, Simplicius's authority, accepted the state- 
ment of Aristotle, without suspecting that the Master might be an 
unsafe guide on such a subject. From the theory of Eudoxus 
as thus restored we can judge how far by his time the Greeks had 
progressed in the study of the motions of the moon. Observations 
had gone far enough to enable the movement in latitude and the 
retrogression of the nodes of the moons orbit to be recognized. 
Eudoxus knew nothing of the variation of the moon's speed in 
longitude, or at least took no account of it, whereas Callippus was 



198 THEORY OF CONCENTRIC SPHERES parti 

aware of it about 325 B.C., that is, about twenty or thirty years 
after Eudoxus's time. 

As regards the sun, we learn from Aristotle that Eudoxus again 
assumed three spheres to explain its motion. As in the case of the 
moon, the first or outermost sphere revolved like the sphere of the 
fixed stars, the second moved about an axis perpendicular to the plane 
of the zodiac, its equator revolving accordingly in the plane of the 
zodiac, while the third moved so that its equator described a plane 
slightly inclined to that of the zodiac, the inclination being less in 
the case of the sun than in the case of the moon. Simplicius adds 
that the third sphere (which is necessary because the sun does not 
at the summer and winter solstices always rise at the same point on 
the horizon) moves much more slowly than the second and (unlike 
the corresponding sphere in the case of the moon) in the direct 
order of the signs. 1 Simplicius makes the same mistake as regards 
the speeds of the second and third spheres as he made in the case 
of the moon. If it were the third sphere which moved very slowly, 
the sun would for ages remain in a north or a south latitude and in 
the course of a year would describe, not a great circle, but (almost) 
a small circle parallel to the ecliptic. The slow motion must there- 
fore belong to the second sphere, the equator of which revolves in 
the ecliptic, while the revolution of the third sphere must take place 
in about a year (strictly speaking, a little more than a tropic year 
in consequence of the supposed slow motion of the second sphere in 
the same sense), the plane of its equator being inclined, at the small 
angle mentioned, to the plane of the ecliptic. The slightly inclined 
great circle of the third sphere which the sun appears to describe 
is then carried round bodily in the revolution of the second sphere 
about the axis of the ecliptic, the nodes on the ecliptic thus moving 
slowly forward, in the direct order of the signs ; and lastly both 
the second and third spheres are carried round by the revolution of 
the first sphere following the daily rotation. 

The strange thing in this description of the sun's motion is the 
imaginary idea that its path is not in the ecliptic but in a circle 
inclined at a small angle to the latter. Simplicius says that 
Eudoxus ' and those who preceded him ' (τοΐ? προ αύτον) thought 
the sun had the three motions described, and that this was inferred 
1 Simplicius on De caelo, pp. 493. 1 5-1 7, 494. 6-7, 9-11. 



ch.xvi EUDOXUS 199 

from the fact that the sun, in the summer and winter solstices, does 
not always rise at the same point of the horizon. 1 We gather from 
this that even before Eudoxus's time astronomers had suspected 
a certain deviation in latitude on the part of the sun. Schiaparelli 
suggests as an explanation that, the early astronomers having dis- 
covered, by comparison with the fixed stars, the deviation of the 
moon and the five planets in latitude, it was natural for them to 
suppose that the sun also must deviate from the circle of the 
ecliptic ; indeed it would be difficult for them to believe that 
the sun alone was exempt from such deviation. However this may 
be, the notion of the nutation of the sun's path survived for centuries. 
Hipparchus 2 quotes a sentence from the lost Enoptron of Eudoxus 
to the effect that ' it appears that the sun too shows a difference in 
the places where it appears at the solstices, though the difference is 
much less noticeable and indeed is quite small'; Hipparchus goes 
on to deny this on the ground that, if it were so, the prophecies by 
astronomers of lunar eclipses, which they made on the assumption 
that there was no deviation of the sun from the ecliptic, would 
sometimes have proved appreciably wrong, whereas in fact the 
eclipses never showed a difference of more than two 'finger- 
breadths ', and only very rarely that, in comparison with the most 
accurately calculated predictions. Notwithstanding Hipparchus's 
great authority, the idea persisted, and we find later authors giving 
a value to the supposed inclination to the ecliptic. We are not told 
what Eudoxus supposed the angle to be, nor what he assumed as 
the period of revolution of the nodes. Pliny 3 gives the inclination 
as i° on each side of the ecliptic ; perhaps he misunderstood his 
source and took a range of i° to be an inclination of i°. For Theon 
of Smyrna, 4 on the authority of Adrastus, says that the inclination 
is J° ; Theon also says that the sun returns to the same latitude 
after 365^ days, whereas it takes 365^ days to return to the same 
equinox or solstice and 365^ days to return to the same dis- 
tance from us. 5 This shows that the solar nodes were thought to 
have a retrograde motion (not a motion in the order of the signs, 

1 Simplicius, loc. cit., p. 493. 11-17. 

2 Hipparchus, In Arati et Eudoxi phaenomena y i. 9, pp. 88-92, ed. Manitius. 

3 Pliny, N.H. ii. c. 16, § 67. 

4 Theon of Smyrna, ed. Hiller, pp. 135. 12-14, 194. 4-8. 
6 Ibid., p. 172. 15-173• 16. 



200 THEORY OF CONCENTRIC SPHERES parti 

as assumed by Eudoxus) and a period of 365%+^ or 2922 years. 
It is not known who invented this theory in the first instance. 
Schiaparelli shows that it was not started for the purpose of 
explaining the motion of the equinoctial points, or the precession 
of the equinoxes, which was discovered by Hipparchus, but was 
unknown to Eudoxus, Pliny, and Theon. 

Eudoxus supposed the annual motion of the sun to be perfectly 
uniform ; he must therefore have deliberately ignored the discovery, 
made by Meton and Euctemon 60 or 70 years before, that the sun 
does not take the same time to describe the four quadrants of its 
orbit between the equinoctial and solstitial points. Eudoxus, in fact, 
seems to have definitely regarded the length of the seasons as being 
as nearly as possible equal, since he made three of them 91 days in 
length, only giving 92 days to the autumn in order to make up 365 
days in the year. 1 

In the case of each of the planets Eudoxus assumed four spheres. 
The first and outermost produced the daily rotation in 24 hours, as 
in the case of the fixed stars ; the second produced the motion along 
the zodiac ' in the respective periods in which the planets appear to 
describe the zodiac circle ', 2 which periods, in the case of the superior 
planets, are respectively equal to the sidereal periods of revolution, 
and in the case of Mercury and Venus (on a geocentric system) one 
year. As the revolution of the second sphere was taken to be 
uniform, we see that Eudoxus had no idea of the zodiacal anomaly 
of the planets, namely that which depends on the eccentricity of 
their paths, and which later astronomers sought to account for by 
the hypothesis of eccentric circles ; for Eudoxus the points on the 
ecliptic where successive oppositions or conjunctions took place were 
always at the same distances, and the arcs of retrogradation were 
constant for each planet and equal at all parts of the ecliptic. Nor 
with him were the orbits of the planets inclined at all to the ecliptic; 

1 This appears from the papyrus known under the title of Ars Eudoxi, 
deciphered by Letronne and published by Brunet de Presle {Notices et extraits 
des tnanuscrits, xviii. 2, 1865, p. 25 sq.). The papyrus was edited by Blass (Kiel, 
1887), and a translation will be found in Tannery's Recherches sur Vhistoire 
de Ρ astronomie ancienne^ pp. 283-94. Tannery prefers the title restored by 
Letronne, Didascalie celeste de Leptine. The document, written in Egypt 
between the years 193 and 165 B.C., seems to have been a student's note-book, 
written perhaps during or after a course of lectures. 

2 Simplicius, loc. cit., p. 495. 25. 



ch.xvi EUDOXUS 201 

their motion in latitude was believed by Eudoxus to depend exclu- 
sively on their elongation from the sun and not on their longitude. 
The third sphere had its poles at two opposite points on the zodiac 
circle, the poles being carried round in the motion of the second 
sphere ; the revolution of the third sphere about the poles was 
again uniform and took place in a period equal to the synodic 
period of the planet or the time which elapsed between two succes- 
sive oppositions or conjunctions with the sun. The poles of the third 
sphere were different for all the planets, except that they were the 
same for Mercury and Venus. The third sphere rotated according 
to Simplicius ' from south to north and from north to south n (this 
followed of course from the position of the poles on the ecliptic) ; 
the actual sense of the rotation is not clear from this, but Schia- 
parelli's exposition shows that it is immaterial whether we take the 
one or the other. On the surface of the third sphere the poles of 
the fourth sphere were fixed, the axis of the latter being inclined to 
that of the former at an angle which was constant for each planet 
but different for the different planets. And the rotation of the 
fourth sphere about its axis took place in the same time as the rota- 
tion of the third about its axis but in the opposite sense. On the 
equator of the fourth sphere the planet was fixed, the planet having 
thus four motions, the daily rotation, the circuit in the zodiac, and 
two other rotations taking place in the synodic period, 

Simplicius gives the following clear explanation with regard to 
the combined effect of the rotations of the third and fourth spheres. 

{ The third sphere, which has its poles on the great circle of the 
second sphere passing through the middle of the signs of the zodiac, 
and which turns from south to north and from north to south, will 
carry round with it the fourth sphere which also has the planet 
attached to it, and will moreover be the cause of the planet's move- 
ment in latitude. But not the third sphere only ; for, so far as it 
was on the third sphere (by itself), the planet would actually have 
arrived at the poles of the zodiac circle and would have come near 
to the poles of the universe ; but, as things are, the fourth sphere, 
which turns about the poles of the inclined circle carrying the 
planet and rotates in the opposite sense to the third, i.e. from east 
to west, but in the same period, will prevent any considerable diver- 
gence (on the part of the planet) from the zodiac circle, and will 

1 Simplicius, loc. cit., p. 496. 23. 



202 THEORY OF CONCENTRIC SPHERES parti 

cause the planet to describe about this same zodiac circle the 
curve called by Eudoxus the hippopede, so that the breadth of this 
curve will be the (maximum) amount of the apparent deviation of the 
planet in latitude, a view for which Eudoxus has been attacked.' x 

Following up the hint here given, Schiaparelli set himself to 
investigate the actual path of a planet subject to the rotations of 
the third and fourth spheres only, leaving out of account for the 
moment the motions of the first two spheres producing respectively 
the daily rotation and the motion along the zodiac. The problem 
is, as he says, in its simplest expression, the following. ' A sphere 
rotates uniformly about the fixed diameter AB. Ρ, P' are two 




Fig. 6. 

opposite poles on this sphere, and a second sphere concentric with 
the first rotates uniformly about PP f in the same time as the former 
sphere takes to turn about A B, but in the opposite direction. Μ is a 
point on the second sphere equidistant from the poles P y P' (in other 
words, a point on the equator of the second sphere). Required to find 
the path of the point M.' This is not difficult nowadays for any 
one familiar with spherical trigonometry and analytical geometry ; 
but it was necessary for Schiaparelli to show that the solution was 
within the powers of Eudoxus. He accordingly develops a solution 
by means of a series of seven propositions or problems involving 
only elementary geometrical considerations, which would have 

1 Simplicius, loc. cit., pp. 496. 23 -497. 5• 



CH. XVI 



EUDOXUS 



203 



presented no difficulty to a geometer of the calibre of Eudoxus ; 
and he finds that, sure enough, the path of Μ in space is a figure 
like a lemniscate but described on the surface of a sphere, that is, the 
fixed sphere about AB as diameter. This ' spherical lemniscate ', 
as Schiaparelli calls it, is shown as well as I can show it in the 
annexed figure (Fig. 7). Its double point is on the circumference of 
the plane section of the sphere which is at right angles to AB, and 
it is symmetrical about that plane as well as about the circumfer- 
ence of a circular section which has AB for diameter and is in what 
Schiaparelli calls the * fundamental plane ', the plane of the great 
circle with diameter A Β on which the pole Ρ and the planet Μ are 




Fig. 7. 

found at the same moment. The curve is actually the intersection 
of the sphere with a certain cylinder touching it internally at the 
double point, namely a cylinder with diameter equal to AS, 
the sagitta (see Fig. 6) of the diameter of the small circle of 
the sphere on which the pole Ρ revolves. But the curve is also 
the intersection of either the sphere or the cylinder with a certain 
cone with vertex O, axis parallel to the axis of the cylinder 
(i.e. touching the circle A OB at O) and vertical angle equal to 
the ' inclination ' (the angle AO'P in Fig. 6). 1 For clearness' sake 

1 Schiaparelli's geometrical propositions are too long to be quoted here, but 
the whole thing can be worked out analytically in a reasonable space. This is 
done by Norbert Herz (Geschichte der Bahnbestimmung von Planeten und 



204 



THEORY OF CONCENTRIC SPHERES parti 



I show in another figure (Fig. 9, p. 206) a right section of the cylinder 
by a plane passing through Ο and perpendicular to AB in the figure 
immediately preceding (Fig. 7). 

The arc of the great circle A OB which bisects the 'spherical 
lemniscate ' laterally is equal in length to the arc QAR of the 
great circle AQBR (Figs. 6 and 8) and is of course divided at the 
double point Ο into equal halves of length equal to the arc AQ. 

Kometen, Part I, Leipzig, 1887, pp. 20, 21), and I quote the solution exactly as 
he gives it : — 

Let AB be the axis of the first sphere, and the circle A OB the circle in 
which P, P' f the poles of the second sphere, and Μ the position of the planet, 
are found together at the same moment. Suppose that the motion of the two 
spheres is in the direction of the arrows and that, when the circle APB has 
moved through an angle θ, PM, the circle carrying the planet has also moved 
through the same angle, Μ being the position of the planet. 




Fig. 8. 

Let i be the inclination AO'P, r the arc of a great circle AM, u (measured 
positively downwards) the angle OAM. 

Then in the triangle Ρ AM we have, since Ρ Μ = 90 , 

cos r = — sin i cos β, 
sin r cos (Θ + u) — + cos i cos Θ, 
sin r sin (Θ + u) = 4- sin Θ. 

Multiplying the second equation by (-sin#) and the third by costf, and 
adding, we have 

sin r sin u = sin # cos θ (ι — cos z) = sin 2 \i sin 2 Θ. 

Multiplying the second equation by cos θ and the third by sin Θ, and adding, 
we have 

sin r cos u = sin 2 θ + cos/ cos 2 θ 

= (cos 2 1 i+ sin 2 ^ i) sin 2 Θ+ (cos 2 J i- sin 2 \ i) cos 2 β 
= cos 2 \i — sin 2 \i cos 2 θ 
= I — 2 sin 2 1 / cos 2 Θ. 



ch.xvi EUDOXUS 205 

The breadth of the ' lemniscate ', i.e. the linear distance between 
the two points on either loop of maximum latitude, north and 
south, is equal to the diameter of the cylinder, i.e. to the 
sagitta AS. The angle at which the curve intersects itself at Ο is 
equal to the inclination {PO'A) of the axes of rotation of the two 
spheres. The four points on the curve of greatest latitude, the 
double point and the two extreme points at which it intersects 

Next, in the triangle A OM, if OM ' — p, and ν is the exterior angle at O, we 
have, since A Ο = go°, 

cos ρ = sin r cos u, 
sin ρ sin ν = sin r sin u, 
sin ρ cos ν — — cos r ; 

therefore, if £, η be the ' spherical coordinates ' of Μ with reference to origin O, 
we have, in the triangle OMN, and by using the results obtained above, 

sin η — sin ν sin ρ = sin 2 \ i sin 2 Θ, 

x „ cot p 1—2 sin 2 1 i cos 2 β 

cot t = = — tan r cos u = : — h* — ^ 

cos ν sin ι cos θ 

If now we use a system of rectangular coordinates x, y, z, with origin at O, 
2 being measured along 00', and x, y being the projections of the arcs ξ, η on 
the plane AQB at right angles to 00' (y being positive in the upward direction, 
i. e. in the opposite direction to u, v), we have for the projections O'N', M'N, 
v' y p of ON, MN> ν, ρ respectively 

O'N' = x\ M'N' = -y } 
v' = v, 
p = 7?sinp, 
where R is the radius of the sphere. 
Consequently χ = ρ cos ν' = R sin ρ cos ν = — R cos r, 

j = — p'sin-z/ = — 7? sin ρ sin 2/ = —Rsinrsmu ; 
whence we have 

χ = /?sin/cos#, 

j = — i? sin 2 ^z' sin 2 0. 

This gives at once the projection of the hippopede on the plane AQB as 
constructed by Schiaparelli. 

So far Norbert Herz. But we can also obtain the remainder of Schiaparelli's 
results, as follows. 

We have for z, the third coordinate of M, 

ζ = R (1 — cosp) = R(i — sin r cos u) 
= 2 R s'm" 1 i cos 2 θ = j?sin 2 £z(l + cos20). 

Eliminating θ from the equations for y and z, we obtain 
(z-R sin 2 1 if +y 2 = R 2 sin 4 * z. 

Therefore J/ lies on a cylinder which has its axis parallel to AB, touches the 
sphere internally at O, and has its radius equal to R sin 2 J- 1, i.e. its diameter 
equal to R (1 - cos i ), which is the sagitta ^ 6" in Fig. 6 and Fig. 8. That is, the 
hippopede is the curve of intersection of this cylinder with the sphere. 

The sphere being x 2 +y 2 + z 2 = 2Rz, and the cylinder y 2 + z 2 = 2 Rz sin 2 \i, 
the cone is easily found to be 

x 2 +y 2 + g* = χ 2 sec 2 \ i. 



1θ6 



THEORY OF CONCENTRIC SPHERES parti 



the 'fundamental plane' through ΛΒ, divide the curve into eight 
arcs which are described by the planet in equal times. Schia- 
parelli shows how to construct the projection of the curve upon the 
plane through AB perpendicular to the plane which bisects 
the curve longitudinally. Describe, he says, a circle with radius 
equal to QS, the radius of the small circle described by Ρ (Fig. 6). 
Then, with the same centre, draw a smaller circle with radius equal 
to half the sagitta AS. Divide the lesser circle into any number of 
equal parts, say 8, as at the points marked ο, i, 2, 3 ... 7 round the 
circle, and suppose the same points marked again with the numbers 
8, 9, 10 . . . 15 respectively ; divide the greater circle into double the 





Fig. 9. 



Fig. 10. 



number of equal parts as at the points marked o, 1, 2, 3, 4, 5 . . . 15 
(arranging the points so that those marked ο are opposite one 
another on a common diameter XX, while the numbers go round 
in the same sense). Draw Υ Υ through the centre perpendicular 
to XX, and through the points of division of the outer circle 
draw chords parallel to YY, and through the points of division 
of the inner circle straight lines parallel to XX. The points of 
intersection of the lines give a series of points on the projection 
of the ' spherical lemniscate '. These points are again marked in 
the figure by the numbers o, 1, 2 .... 15. The projection of the 
position of the planet moves along this curve in the direction indi- 
cated by the successive numbers. 



ch.xvi EUDOXUS 207 

There is no doubt that Schiaparelli has restored, in his c spherical 
lemniscate ', the hippopede of Eudoxus, the fact being confirmed by 
the application of the same term hippopede (horse-fetter) by Proclus 1 
to a plane curve of similar shape formed by a plane section of an 
anchor-ring or tore touching the tore internally and parallel to its 
axis. 

So far account has only been taken of the motion due to the com- 
bination of the rotations of the third and fourth spheres. But A, B, 
the poles of the third sphere (Figs. 6-8), are carried round the zodiac 
or ecliptic by the motion of the second sphere and in a time equal to 
the ' zodiacal ' period of the planet. Now the longitudinal axis of 
the ' spherical lemniscate ' (the arc of the great circle bisecting it 
longitudinally) always lies on the ecliptic. We may therefore sub- 
stitute the ' lemniscate ' moving bodily round the ecliptic for the 
third and fourth spheres, the planet meantime moving round 
the ' lemniscate ' in the manner described above. The combination 
of the two motions (that of the ' lemniscate ' and that of the planet 
on it) gives the motion of the planet through the constellations. 
The motion of the planet round the curve is an oscillatory motion, 
now forward in acceleration of the motion round the ecliptic due to 
the second sphere, now backward in retardation of the same motion ; 
the period of the oscillation is the period of the synodic revolution, 
and the acceleration and retardation occupy half the period respec- 
tively. When the retardation in the sense of longitude due to the 
backward oscillation is greater than the speed of the forward motion 
of the i lemniscate ' itself, the planet will for a time have a retrograde 
motion, at the beginning and end of which it will appear stationary 
for a little while, when the two opposite motions balance each 
other. The greatest acceleration of the planet in longitude, and the 
greatest retardation (or the quickest rate of retrograde motion), 
occur at the times when the planet passes through the double point 
of the curve. The movements must therefore be so combined that 
the planet is at the double point and moving in the forward direction 
at the time of superior conjunction with the sun, when the apparent 
speed of the planet in longitude is greatest, while it is again at 
the double point but moving in the backward direction when it 
is in opposition or inferior conjunction, at which times the apparent 
1 Proclus, Comm. on Eucl. I, ed. Friedlein, p. 112. 5. 



2o8 THEORY OF CONCENTRIC SPHERES parti 

retrograde motion of the planet is quickest. This combination of 
motions will be accompanied by motion in latitude within limits 
defined by the breadth of the lemniscate ; the planet will, during 
a synodic revolution, twice reach its greatest north and south lati- 
tude respectively and four times cross the ecliptic. 

The actual shape of the hippopede and its dimensions relatively to 
the sphere on which it is drawn are fully determined when we know 
the inclination of the axis of the fourth sphere to that of the third, 
since they depend on this inclination exclusively. In order to test 
the working of the theory with regard to the several planets we 
need to know three things, (i) the inclination referred to, (a) the 
period of the ' zodiacal ' or sidereal revolution, (3) the synodic 
period, in the case of each planet. We are not told what angles of 
inclination Eudoxus assumed, but the zodiacal and synodic periods 
which he ascribed to the five planets are given in round figures by 
Simplicius. 1 The following is a comparison of Eudoxus's figures 
with the modern values : 

Synodic period Zodiacal period 



Eudoxus Modern value Eudoxus Modern value 

Saturn 13 months 378 days 30 years 29 years 166 days 

Jupiter 13 months 399 days 12 years 11 years 315 days 

Mars 8 months 20 days 780 days 2 years 1 year 322 days 

Mercury no days 116 days 1 year 1 year 

Venus 19 months 584 days I year I year 

Except in the case of Mars, these figures are tolerably accurate, 
while the papyrus purporting to contain the Ars Eudoxi gives for 
the synodic period of Mercury the exact modern figure of 116 days ; 
it is therefore clear that Eudoxus went on the basis of very careful 
observations, whether he obtained the results from Egypt or from 
Babylonian sources. As unfortunately the inclinations assumed by 
Eudoxus (the third factor required for the reconstruction of the 
system) are not recorded, Schiaparelli has to conjecture them for 
himself. Assuming that they would be such as to produce ' lem- 
niscates ' which would give arcs of retrogradation corresponding to 
those actually observed, he takes the known retrograde arc of 
Saturn (6°) and observes that by the help of the zodiacal period 
of 30 years and the synodic period of 13 months, and by assuming 

1 Simplicius, loc. cit., pp. 495. 26-9, 496. 6-9. 



CH. xvi EUDOXUS 209 

6° as the ' inclination ', a retrograde arc of about 6° is actually 
obtained ; the length of the hippopede (the arc of the great circle of 
the sphere bisecting the curve longitudinally) is 1 2°, and the half of 
its breadth about 9/, a maximum deviation from the ecliptic which 
would of course be imperceptible to the observers of those days. 
In the case of Jupiter, assuming an inclination of 13°, and conse- 
quently a hippopede of 26° in length and twice 44/ in breadth, 
with a zodiacal period of 12 years and a synodic period of 13 
months, he deduces a retrograde arc of about 8° ; and again the 
divergence in latitude of 44/ would hardly be noticed. For these 
two planets, therefore, Eudoxus's method gave an excellent solution 
of Plato's problem of finding how the motion of the planets can be 
accounted for by a combination of uniform circular motions. 

With Mars, however, the system fails. We have no means of 
knowing how Eudoxus came to put the synodic period at 8 months 
and 20 days, or 260 days, whereas it is really 780 days, or three 
times as long. But, whether we take 780 days or 260 days, the 
theory does not account for the facts. If the synodic period is 780 
days, and we take for the length of the hippopede the greatest arc 
permissible according to Simplicius's account, namely an arc of 
180 , corresponding to an 'inclination' of 90 , the breadth of the 
curve becomes 6o°, so that Mars ought to diverge in latitude to 
the extent of 30 . Also, even under this extreme hypothesis, the 
retrograde motion of Mars on the hippopede cannot reach a speed 
equal to that of the direct motion of the hippopede itself along the 
ecliptic (the zodiacal period being 2 years) ; consequently Mars 
should not have any retrograde motion at all and should only move 
very slowly at opposition. To obtain a retrograde motion at all 
we should require an ' inclination ' greater than 90 , and consequently 
the third and fourth spheres would rotate in the same instead of the 
opposite sense, which is contrary to Simplicius's statement; and, 
even if this were permissible, there is the objection that Mars's 
deviations in latitude would exceed 30 , and Eudoxus would never 
have assumed such an amount of deviation. On the other hand, to 
assume a synodic period of 260 days would produce a retrograde 
motion ; by assuming an inclination of 34° we get 68° as the length 
of the hippopede and a maximum deviation in latitude of 4 $$\ 
which is not very far from the true deviation ; the retrograde arc 

1410 Ρ 



aio THEORY OF CONCENTRIC SPHERES parti 

becomes about i6°, which is little greater than that disclosed by 
observation. This way of producing approximate agreement with 
observed facts may perhaps have been what led Eudoxus to assume 
a synodic period one third as long as the real period ; but unfor- 
tunately the hypothesis gives two retrograde motions outside the 
oppositions with the sun, and six stationary points, four of which 
have no real existence. 

As regards Mercury and Venus, inasmuch as their mean positions 
coincide with the mean position of the sun, Eudoxus must have 
assumed that the centre of the hippopede always coincides with 
the sun. This centre being on the ecliptic and at a distance of 
90 from each of the poles of rotation of the third sphere, the poles 
of the third sphere of Mercury and the poles of the third sphere 
of Venus coincide, a fact for which we have the independent 
testimony of Aristotle in the passage quoted above. As the mean 
position of each of the two planets coincides with that of the sun, 
and the greatest elongation of each from the sun is half the length 
of the corresponding hippopede, Eudoxus doubtless determined 
the 'inclination' from the observed elongations. In the case of 
Mercury, with a maximum elongation of 23°, the length of the 
hippopede becomes 46 , and the half of its breadth or the greatest 
latitude is 2° 14', nearly as great as the observed deviation. The 
retrograde arc for Mercury would be about 6°, which is much 
smaller than the true length ; but, as this mistake occurs in a part 
of the synodic circuit which cannot be observed, the theory cannot 
be blamed for this. In the visible portions of the circuit the 
longitudes are represented with fair accuracy, though the times 
of greatest elongation do not exactly agree with the facts. For 
Venus, taking the greatest elongation (and consequently the ' in- 
clination') at 46°, we have a hippopede 92° in length, and a half- 
breadth or maximum latitude of 8° 54', which is roughly in agreement 
with the greatest latitude as observed. But, since the synodic period 
as given by Eudoxus, 570 days, is more than \\ times the zodiacal 
period, Venus, like Mars, can never have a retrograde motion ; 
and this error cannot be avoided whatever value we choose to 
substitute for 46 as the inclination. Another serious fault of the 
theory is that it requires Venus to take the same time to pass 
from the extreme eastern point to the extreme western point 



ch.xvi EUDOXUS 211 

of the hippopcde as it takes to return from the extreme 
western to the extreme eastern point, whereas in fact Venus takes 
441 days (out of the synodic period of 584 days) to pass from the 
greatest eastern to the greatest western elongation and only 143 
days to return from the greatest western to the greatest eastern 
elongation. As regards latitude, too, the imperfection of the theory 
is more marked in the case of Venus than in that of the other 
planets ; for the hippopede intersects the ecliptic four times, once 
at each extremity, and twice at the double point ; consequently 
the planet ought to cross the ecliptic four times during each synodic 
period, which is not the case, as the latitude of Venus is only nil 
twice during each sidereal revolution. 

To sum up. For the sun and moon the hypothesis of Eudoxus 
sufficed to explain adequately enough the principal phenomena, 
except the irregularities due to the eccentricities, which were either 
unknown to Eudoxus or neglected by him. For Jupiter and 
Saturn, and to some extent for Mercury also, the system was 
capable of giving on the whole a satisfactory explanation of their 
motion in longitude, their stationary points and their retrograde 
motions ; for Venus it was unsatisfactory, and it failed altogether 
in the case of Mars. The limits of motion in latitude represented 
by the various hippopedes were in tolerable agreement with observed 
facts, although the periods of the deviations and their places in 
the cycle were quite wrong. But, notwithstanding the imper- 
fections of the system of homocentric spheres, we cannot but 
recognize in it a speculative achievement which was worthy of the 
great reputation of Eudoxus and all the more deserving of admira- 
tion because it was the first attempt at a scientific explanation of 
the apparent irregularities of the motions of the planets. And, 
as Schiaparelli says, if any one, as the result of a superficial study 
of the theory, finds it complicated, let him remember that in 
none of his hypotheses does Eudoxus make use of more than three 
constants or elements, namely the epoch of a superior conjunction, 
the period of sidereal revolution (on which the synodic period is 
dependent), and the inclination to one another of the axes of the 
third and fourth spheres, which inclination determines completely 
the dimensions of the hippopede ; whereas in our time we require, for 
the same purpose, six elements in the case of each planet. 

Ρ 2 



212 THEORY OF CONCENTRIC SPHERES parti 

Eudoxus died in 355 B. c. at the age of 53. His doctrine of 
homocentric spheres was further studied in his school. Menaechmus, 
the reputed discoverer of the conic sections, and one of his pupils, 
is mentioned as a supporter of the theory. 1 Polemarchus of 
Cyzicus, a friend of Eudoxus, is also mentioned as having studied 
the subject, and, in particular, as having been aware of the objection 
raised to the system of homocentric spheres on the ground that 
the difference in the brightness of the planets, especially Venus and 
Mars, and in the apparent size of the moon, at different times, 
showed that they could not always be at the same distance from 
us ; * Polemarchus appears to have been aware of it ' (the variation 
in the distances of each planet) ' but to have neglected it as not per- 
ceptible, because he preferred the assumption that the spheres them- 
selves are about the centre of the universe \ 2 But it is Callippus 
to whom definite improvements in the system are attributed. 
Callippus of Cyzicus, the most famous and capable astronomer of 
his time, probably lived between 370 and 300 B.C. ; he was therefore 
perhaps too young to be a pupil of Eudoxus himself; but he studied 
with Polemarchus and he followed 3 Polemarchus to Athens, where 
' he stayed with Aristotle correcting and completing, with Aristotle's 
help, the discoveries of Eudoxus'. 4 This must have been during 
the reign of Alexander the Great (336-323 B.C.), at which time 
Aristotle was in Athens ; it must also have been about the time 
when Callippus brought out his improvement of Meton's luni-solar 
cycle, since the beginning of Callippus's cycle was in 330 B. C. 
(28th or 29th June). Aristotle himself gives Callippus the sole 
credit for certain improvements on Eudoxus's system ; immediately 
after the passage above quoted from the Metaphysics he says : 

1 Callippus agreed with Eudoxus in the position he assigned to 
the spheres, that is to say, in their arrangement in respect of 
distances, and he also assigned the same number of spheres as 
Eudoxus did to Zeus and Kronos respectively, but he thought it 
necessary to add two more spheres in each case to the sun and 
moon respectively, if one wishes to account for the phenomena, 
and one more to each of the other planets.' 5 

1 Theon of Smyrna, ed. Hiller, pp. 201. 25-202. I. 

2 Simplicius on De caelo ii. 12, p. 505. 21, Heib. 

3 μίτ ζκΐίνον els Αθήνας έλθών. Schiaparelli translates /xer intivov as if it were 
μςτ ('κείνου, * with him \ 

4 Simplicius, op. cit., p. 493. 5-8. 6 Aristotle, Metaph. Λ. 8, 1073 b 32-8. 



ch. xvi CALLIPPUS 213 

Simplicius says that no book by Callippus on the subject was extant 
in his time, nor did Aristotle give any explanation of the reasons why 
Callippus added the extra spheres ; 

1 but Eudemus shortly stated what were the phenomena in explana- 
tion of which Callippus thought it necessary to assume the additional 
spheres. According to Eudemus, Callippus asserted that, assuming 
the periods between the solstices and equinoxes to differ to the 
extent that Euctemon and Meton held that they did, the three 
spheres in each case (i. e. for the sun and moon) are not sufficient 
to save the phenomena, in view of the irregularity which is observed 
in their motions. But the reason why he added the one sphere 
which he added in the case of each of the three planets Ares, 
Aphrodite, and Hermes was shortly and clearly stated by Eudemus.' x 

As regards the planets therefore, although we are informed that 
Eudemus gave the reason for the addition of a fifth sphere in each 
case, we are not told what the reason was, and we can only resort 
to conjecture. Schiaparelli observes that, since Callippus was 
content with Eudoxus's hypothesis about Jupiter and Saturn, we 
may conclude that their zodiacal inequality was still unknown to 
him, although it can reach the value of 6° in each case, and also 
that he regarded their deviations in latitude as non-existent or 
negligible. But the glaring deficiencies in the theory of Eudoxus 
when applied to Mars would suggest the urgent need for some 
improvement which should, in particular, produce the necessary 
retrograde motion in this case without the assumption of a synodic 
period different from the true one. It is sufficiently probable there- 
fore that the fifth sphere was intended for the purpose of satisfying 
this latter condition. Schiaparelli observes that, on the assumption 
of a synodic period of 780 days, it is possible, by a combination 
of three spheres taking the place of Eudoxus's last two (the third 
and fourth), to obtain a retrograde motion agreeing sufficiently with 
observed facts, and this can be done in various ways without 
involving too considerable deviations in latitude ; he gives, as the 
simplest arrangement leading to the desired result, the following. 

Let A OB (Fig. 11) represent the ecliptic and A, Β two opposite 
points on it which make the circuit of the zodiac in the zodiacal 
period of Mars. Let a sphere (the third of Eudoxus) revolve about 

1 Simplicius on De caelo ii. 12, p. 497. 17-24. 



214 



THEORY OF CONCENTRIC SPHERES parti 



A y Β as poles in the period of the synodic revolution. Take any 
point Ρ τ on the equator of this sphere as pole of another sphere (the 
fourth) rotating about its poles at twice the speed of the third sphere, 
in the opposite direction to the latter, and carrying with it P 2 , 
distant from P x by an arc P X P 2 (which we will call the ' inclination '). 
About P 2 as pole, let a fifth sphere rotate at the same speed and 




Fig. ii. 

in the same direction as the third, carrying the planet fixed on 
its equator at the point M. It is easy to see that, if at the 
beginning of the motion the three points P Xi P 21 Μ lie on the 
ecliptic in the order ΑΡ 2 Ρ τ ΜΒ } then at any time afterwards 
the angle φ at A will be equal to the angle at P 2 between P 2 P Y and 
P 2 M } while the angle AP 1 P 2 at P x will be twice as large. And, 
since AP Y — P 2 M ' = 90 , the planet Μ will in the synodic period 
describe a curve adjacent to the ecliptic and symmetrical about it 
which will take a different form according to the value given to the 
'inclination' P 1 P 2 . This curve will for certain values of Ρ λ Ρ 2 
extend considerably in length but little in breadth and, as it has 
its centre at Ο midway between the poles A y B, it will, like the 
Ziippopede, produce a direct and retrograde motion alternately, but 



ch.xvi CALLIPPUS 215 

will have the advantage over the hippopede that it can give the 
planet in the neighbourhood of Ο a much greater direct and retro- 
grade velocity with the same motion in latitude. Hence it is 
capable of giving the planet a retrograde motion where the hippopede 
fails to do so. If, for example, P X P 2 is put equal to 45 , the curve 
takes a form like that shown in the figure (in projection). The 
greatest deviation in latitude does not exceed 4 1 i', the curve has 
a length along the ecliptic of 95§°, and has two triple points near 
the ends at a distance of 45 from the centre O. When the planet 
is passing O, its velocity is 1-2929 times the speed of the rotation 
of P x about AB. As the period of rotation of P 1 about AB is 
equal to the synodic period, 780 days, the daily motion of P x 
is 3όΌ°/78ο or ο°•402, which multiplied by 1-2929 gives o°-$9J as 
the daily retrograde motion on the curve at O. And, as Ο has 
a direct motion on the ecliptic of 360 / 686 = o°-525, the resulting 
daily retrograde motion is o -o72, which is a reasonable approxima- 
tion to the fact. 

Similarly an additional sphere might be made to remove the 
imperfection of the theory as applied to Venus. If the ' inclination ' 
P X P 2 is made 45°, the greatest elongation is 47§°, which is very near 
the truth ; and the different speed of the planet in the four parts of 
the synodic revolution is also better accounted for, since, in the 
curve above drawn, the passage from one triple point to the other 
takes one fourth of the time, the same passage back again another 
fourth, while the remaining two fourths are occupied by the very 
slow motion round the small loops at the ends. For Mercury the 
theory of Eudoxus gave a fairly correct result, and doubtless it 
would be possible by means of another sphere to attain a still greater 
degree of accuracy. 

According to Eudemus, Callippus added two new spheres (making 
five) in the case of the sun, in order to account for the unequal 
motion in longitude which had been discovered a hundred years 
earlier by Meton and Euctemon. Euctemon had made the length 
of the seasons (beginning with the vernal equinox) 93,90, 90, and 92 
days respectively, showing errors ranging from 1-23 to 2-01 days ; 
this was about 430 B.C. Callippus, about 330 B.C., made the cor- 
responding lengths 94, 92, 89, 90 days respectively, 1 the errors 

1 Ars Eudoxi, § 55. 



2i6 THEORY OF CONCENTRIC SPHERES parti 

ranging from 008 to 0-44 days only ; this shows the great advance 
made in observations of the sun during the century between the two 
dates. Now Callippus had only to retain the three spheres assumed 
by Eudoxus for the sun and then to add two spheres, (1) a sphere 
with its poles on the third sphere of Eudoxus which described the 
orbit of the sun at uniform speed in the course of a year, and (2) a 
sphere carrying the sun on its equator and having its poles on the 
preceding sphere and its axis slightly inclined to the axis of 
the same sphere ; the second of these spheres would rotate at the 
same speed as the first but in the opposite direction. If the inclina- 
tion of the axes is equal to the greatest inequality (which was for 
Callippus, as it is for us, 2°), the two new spheres give for the sun a 
hippopede, the length of which along the ecliptic is 4 and the breadth 
nearly i' on each side of it ; this representation of the motion of 
the sun is almost as accurate as that obtained later by means of the 
eccentric circle and the epicycle. 

Simplicius's explanation of the reason why Callippus added two 
spheres in the case of the moon also is rather confused, because he 
tries to deal with the sun and moon in one sentence. But he pre- 
sumably meant that the reason in the case of the moon was similar 
to the reason in the case of the sun ; in other words, Callippus was 
aware of the inequality in the motion of the moon in longitude. 
This inequality, which often reaches as much as 8°, would neces- 
sarily reveal itself as soon as the intervals between a large number 
of successive lunar eclipses were noted and compared with the 
corresponding longitudes of the moon, which can in this case easily 
be deduced from those of the sun. The inclination between the 
axes of the two new spheres would in this case have to be taken 
equal to the mean inequality of 6°, and a hippopede of 12 would 
mean a maximum deviation from the moon's path of 9/, so that 
the moon's motion in latitude would not be sensibly affected. 

Whether Callippus actually arranged his additional spheres in 
the way suggested by Schiaparelli or not, the improvements which 
he made were doubtless of the nature indicated above ; and his 
motive was that of better ' saving the phenomena ', his comparison 
of the theory of Eudoxus with the results of actual observation 
having revealed differences sufficiently pronounced to necessitate a 
remodelling of the theory. 



ch.xvi ARISTOTLE 217 

We now come to the changes which Aristotle thought it neces- 
sary to make in the system of Eudoxus and Callippus. We have 
seen that that system was purely geometrical and theoretical ; 
there was nothing mechanical about it. Aristotle's point of view 
was entirely different. Aristotle, as we shall see, transformed the 
purely abstract and geometrical theory into a mechanical system 
of spheres, i.e. spherical shells, in actual contact with one another; 
this made it almost necessary, instead of assuming separate sets of 
spheres, one set for each planet, to make all the sets part of one 
continuous system of spheres. For this purpose yet other spheres 
had to be added which Aristotle calls ' unrolling ' or ' back-rolling ' 
(άι/βλίττονσαι), 1 by which is meant 'reacting' in the sense of counter- 
acting the motion of certain of Eudoxus's and Callippus's spheres 
which, for the sake of distinction, we may with Schiaparelli call 
' deferent '. Aristotle's theory and its motive are given quite clearly 
in the chapter of the Metaphysics to which reference has already 
been made. The words come immediately after the description of 
Callippus's additions to the theory. 

' But it is necessary, if the phenomena are to be produced by all 
the spheres acting in combination {σνντζβύσαι)^ to assume in the 
case of each of the planets other spheres fewer by one [than 
the spheres assigned to it by Eudoxus and Callippus] ; these latter 
spheres are those which unroll, or react on, the others in such a way 
as to replace the first sphere of the next lower planet in the same 
position [as if the spheres assigned to the respective planets above 
it did not exist], for only in this way is it possible for a combined 
system to produce the motion of the planets. Now the deferent 
spheres are, first, eight [for Saturn and Jupiter], then twenty-five 
more [for the sun, the moon, and the three other planets] ; and of 
these only the last set [of five] which carry the planet placed lowest 
[the moon] do not require any reacting spheres. Thus the reacting 
spheres for the first two bodies will be six, and for the next four will 
be sixteen ; and the total number of spheres, including the deferent 
spheres and those which react on them, will be fifty-five. If, how- 
ever, we choose not to add to the sun and moon the [additional 
deferent] spheres we mentioned [i. e. the two added to each by 
Callippus], the total number of the spheres will be forty-seven. So 
much for the number of the spheres.' 2 

1 Theophrastus, we are told (Simplicius, loc. cit., p. 504. 6), called them 
άνταναφερουσαι. 

2 Aristotle, Metaph. Λ. 8, 1073 b 38 - 1074 a 1 5. 



2i8 THEORY OF CONCENTRIC SPHERES parti 

The way in which the system would work is explained very 
diffusely by Sosigenes in Simplicius; 1 Schiaparelli puts the matter 
quite clearly and shortly, thus. The different sets of spheres being 
merged into one, it is necessary to provide against the motion of 
the spheres assigned to a higher planet affecting the motion of the 
spheres assigned to a lower planet. For this purpose Aristotle 
interpolated between the last (the innermost) sphere of each planet 
and the first (or outermost) sphere of the planet next below it 
a certain number of spheres called ' reacting ' spheres. Thus, sup- 
pose A, B, C, D to be the four spheres postulated for Saturn, A 
being the outermost and D the innermost on which the planet is 
fixed. If inside the sphere D we place a first reacting sphere D' 
which turns about the poles of D with equal speed, but in the 
opposite sense, to D, the rotations of D and D' will mutually 
cancel each other and any point of D' will move as though 
it was rigidly connected with the sphere C. Again, if we place 
inside the sphere D' a second reagent sphere C rotating about 
the same poles with C and with equal speed, but in the opposite 
sense, the rotations of C and C' cancel each other, and any point 
of C' will move as if it were rigidly connected with the sphere B. 
Lastly, if inside C a third reagent sphere B' is introduced which 
rotates about the same poles with Β and at the same speed but in 
the opposite sense, the rotations of Β and B f will cancel each other 
and any point of B' will move as if it were rigidly connected with 
the sphere A. But, as A is the outermost sphere for Saturn, A is the 
motion of the sphere of the fixed stars; hence B' will move in 
the same way as the sphere of the fixed stars ; and consequently 
Jupiter's spheres can move inside B' as if the spheres of Saturn did 
not exist and as if B' itself were the sphere of the fixed stars. 

Hence it is clear that, if η is the number of the deferent spheres 
of a planet, the addition of η — ι reacting spheres inside them 
neutralizes the operation of η — ι of the original η spheres and pre- 
vents the inner set of spheres from being disturbed by the outer 
set. The innermost of the ;z — ι reacting spheres moves, as above 
shown, in the same way as the sphere of the fixed stars. But the 
first sphere of the next nearer planet (as of all the planets) is also a 
sphere with the same motion as that of the sphere of the fixed 
1 Simplicius on De caelo ii. 12, pp. 498. I - 503. 9. 



ch.xvi ARISTOTLE 219 

stars, and consequently we have two spheres, one just inside the 
other, with one and the same motion, that is, doing the work of one 
sphere only. Aristotle could therefore have dispensed with the 
second of these, namely the first of the spheres belonging to 
the inner planet, without detriment to the working of his system ; 
and, as the number of ' planets ' inside the outermost, Saturn, is six, 
he could have saved six spheres out of his total number. 

Aristotle omits, as unnecessary, any reacting spheres for the 
last and innermost planet, the moon. Yet, as Martin points out, 1 
Aristotle should have realized that, strictly speaking, the account 
which he gives in the Meteorologica of shooting stars, comets, and 
the Milky Way necessitates the introduction of four reacting 
spheres below the moon. For, according to Aristotle, these 
phenomena are the effects of exhalations rising to the top of the 
sublunary sphere and there coming into contact with another 
warm and dry substance which, being the last layer of the sublunary 
sphere and in contact with the revolution of the outer heavenly 
sphere, is carried round with it ; the rising exhalations are kindled 
by meeting and being caught in the other substance and are carried 
round with it. Hence there must be a sphere below the moon 
which has the same revolution as that of the sphere of the fixed 
stars, in order that comets, &c, may be produced and move as they 
are said to do. The four inner spheres producing the moon's own 
motion should therefore be neutralized as usual by the same 
number of reacting spheres. 

As it is, however, the hypotheses of Callippus, with the additions 
of spheres actually made by Aristotle, work out thus : 

Deferent spheres Reacting spheres 

For Saturn 4 3 

Jupiter 4 3 

Mars 5 4 

Mercury 5 4 

Venus 5 4 

Sun 5 4 

Moon 5 ο 

Total 33 4- 22 = 55 

In saying that, if Callippus's additional spheres for the sun and 
moon are left out, the total number of spheres becomes 47, it would 

1 Menioires de Γ Acad, des Inscr. et Belles~Lettres t xxx. 1881, pp. 263-4. 



22o THEORY OF CONCENTRIC SPHERES parti 

seem that Aristotle made an arithmetical slip ; x for the omission 
would reduce the number 55 by 6 (4 deferent and 1 reacting), not 
by 8, and would leave 49, not 47. The remark would also seem 
to show that Aristotle did not feel quite certain t that the two 
additional spheres assumed by Callippus for the sun and moon 
respectively were really necessary. We may compare the passage 
in the De caelo where he definitely regards the sun and moon as 
having fewer motions than some of the planets ; in that passage 
he endeavours to explain two ' difficulties ' (anopicu), one of which 
is stated as follows : 

( What can be the reason why the principle that the bodies which 
are at a greater distance from the first motion [the daily rotation 
of the sphere of the fixed stars] are moved by more movements 
does not apply throughout, but it is the middle bodies which have 
most movements? For it would appear reasonable that, as the 
first body [the sphere of the fixed stars] has one motion only, 
the nearest body to it should be moved by the next fewest move- 
ments, say two, the next to that by three, or in accordance with 
some other similar arrangement. But in practice the opposite is 
what happens ; for the sun and moon are moved by fewer move- 
ments than some of the planets, and yet the latter are further from 
the centre and nearer the first body [the sphere of the fixed stars] 
than the sun and moon are. In the case of some planets this is 
even observable by the eye ; for, at a time when the moon was 
halved, we have seen the star of Ares go behind it and become 
hidden by the dark portion of the moon and then come out at the 
bright side of it. And the Egyptians and Babylonians of old 
whose observations go back a great many years, and from whom 
we have a number of accepted facts relating to each of the stars, 
tell us of similar occultations of the other stars.' 2 

1 There are other explanations, but they are all somewhat forced, and involve 
greater difficulties than they remove (see Simplicius on De caelo, pp. 503. 10- 
504. 3, and Martin, loc. cit., pp. 265-6). A further reduction of the number 49 
to 47 which might have been, but obviously was not, made by Aristotle, is 
indicated by Martin (loc. cit., p. 268) and by Dreyer {Planetary Systems, 
p. 114 note). Aristotle might have abolished the i third'' of the sun's spheres 
(as well as the fourth and fifth) ; this would have been a real improvement, 
since the ' third ' sphere was meant to explain a movement which did not exist, 
namely, the supposed movement of the sun in latitude ; the number of the 
spheres would thus have been reduced by two (one deferent and one reacting). 
But Aristotle had not the knowledge necessary to enable him to suggest this 
improvement. 

2 De caelo ii. 12, 291 b 29 - 292 a 9. 



ch.xvi ARISTOTLE 221 

Aristotle's explanation is teleological, based on comparison with 
things which have life and are capable of action. We may perhaps 
say that that thing is in the best state which possesses the good 
without having to act at all, while those come nearest the best 
state which have to perform the fewest acts. 1 Now the earth is 
in the most happy state, being altogether without motion. The 
bodies nearest to it have few movements ; they do not attain the 
ideal, but come as near as they can to ' the most divine principle \ 
The ' first heaven ' [the sphere of the fixed stars] attains it at once 
by means of one movement only ; the bodies between the first and 
the last [the last being the sun and the moon] attain it but only by 
means of a greater number of movements. 2 — 

The theory of concentric spheres was pursued for some time after 
Aristotle. Schiaparelli conjectures that even Archimedes still held 
to it. Autolycus, the author of the treatises On the moving sphere 
and On risings and settings, who lived till the end of the fourth 
or the beginning of the third century B.C., is said to have been 
the first to try, presumably by some modification of the theory, 
to meet the difficulties which had been seen from the first and 
were doubtless pointed out with greater insistence as time went on. 
What was ultimately fatal to it was of course the impossibility 
of reconciling the assumption of the invariability of the distance 
of each planet with the observed differences in the brightness, 
especially of Mars and Venus, at different times, and the apparent 
difference in the relative sizes of the sun and moon. The quotation 
by Simplicius from Sosigenes on this subject is worth giving in full. 4 

'Nevertheless the theories of Eudoxus and his followers fail to 
save the phenomena, and not only those which were first noticed 
at a later date, but even those which were before known and 
actually accepted by the authors themselves. What need is there 
for me to mention the generality of these, some of which, after 
Eudoxus had failed to account for them, Callippus tried to save, — 
if indeed we can regard him as so far successful ? I confine my- 
self to one fact which is actually evident to the eye ; this fact 
no one before Autolycus of Pitane even tried to explain by means 
of hypotheses (Sloc των υποθέσεων), and not even Autolycus was 
able to do so, as clearly appears from his controversy with 

1 De caelo ii. 12, 292 a 22-4. 2 Ibid. 292 b 10-25. 

3 Simplicius on De caelo, pp. 504. 17-505. 11, 505. 19-506. 3. 



222 THEORY OF CONCENTRIC SPHERES parti 

Aristotherus 1 . I refer to the fact that the planets appear at 
times to be near to us and at times to have receded. This is 
indeed obvious to our eyes in the case of some of them ; for the 
star called after Aphrodite and also the star of Ares seem, in 
the middle of their retrogradations, to be many times as large, so 
much so that the star of Aphrodite actually makes bodies cast 
shadows on moonless nights. The moon also, even in the perception 
of our eye, is clearly not always at the same distance from us, because 
it does not always seem to be the same size under the same 
conditions as to medium. The same fact is moreover confirmed 
if we observe the moon by means of an instrument ; for it is at 
one time a disc of eleven fingerbreadths, and again at another 
time a disc of twelve fingerbreadths, which when placed at the 
same distance from the observer hides the moon (exactly) so that 
his eye does not see it. In addition to this, there is evidence for 
the truth of what I have stated in the observed facts with regard 
to total eclipses of the sun ; for when the centre of the sun, the 
centre of the moon, and our eye happen to be in a straight line, 
what is seen is not always alike ; but at one time the cone which 
comprehends the moon and has its vertex at our eye comprehends 
the sun itself at the same time, and the sun even remains invisible 
to us for a certain time, while again at another time this is so far 
from being the case that a rim of a certain breadth on the outside 
edge is left visible all round it at the middle of the duration of the 
eclipse. Hence we must conclude that the apparent difference in 
the sizes of the two bodies observed under the same atmospheric 
conditions is due to the inequality of their distances (at different 
times). . . . But indeed this inequality in the distances of each star 
at different times cannot even be said to have been unknown to 
the authors of the concentric theory themselves. For Polemarchus 
of Cyzicus appears to be aware of it, but to minimize it as being 
imperceptible, because he preferred the theory which placed the 
spheres themselves about the very centre in the universe. Aristotle 
too, shows that he is conscious of it when, in the Physical Problems, 
he discusses objections to the hypotheses of astronomers arising 
from the fact that even the sizes of the planets do not appear to be 
the same always. In this respect Aristotle was not altogether 
satisfied with the revolving spheres, although the supposition that, 
being concentric with the universe, they move about its centre 
attracted him. Again, it is clear from what he says in Book Λ 
of the Metaphysics that he thought that the facts about the move- 
ments of the planets had not been sufficiently explained by the 

1 Apparently a contemporary of Autolycus and, like him, a mathematician. 
The famous poet Aratus appears to have been a pupil of Aristotherus (Buhle's 
Aratus, Leipzig, 1793, vol. i, p. 4). 



ch.xvi EARLY CRITICISMS 223 

astronomers who came before him or were contemporary with him. 
At all events we find him using language of this sort : " (on the ques- 
tion how many in number these movements of the planets are), we 
must for the present content ourselves with repeating what some 
of the mathematicians say, in order that we may form a notion and 
our mind may have a certain definite number to apprehend ; but 
for the rest we must investigate some matters for ourselves and 
learn others from other investigators, and, if those who study these 
questions reach conclusions different from the views now put forward, 
we must, while respecting both, give our adherence to those which 
are the more correct " ', 1 

Schiaparelli observes that we must not be misled by these 
attempts to father on Aristotle doubts as to the truth of the theory 
of homocentric spheres; the object is to make an excuse for the 
line taken by the later Peripatetics in getting away from the 
revolving spheres of Aristotle and going over to the theory of 
eccentric circles and epicycles. 

The allusion by Sosigenes to annular eclipses of the sun is 
particularly interesting, as it shows that he had much more correct 
notions on this subject than most astronomers up to Tycho Brahe. 
Even at the beginning of the seventeenth century, says Schiaparelli, 
some persons doubted the possibility of a total eclipse. Proclus 
points out that the views of Sosigenes are inconsistent with the 
opinion of Ptolemy that the apparent diameter of the sun is always 
the same, while that of the moon varies and is only at its apogee 
the same as that of the sun. * If the latter contention is true/ 
says Proclus, 2 'then that is not true which Sosigenes said in his 
work On the revolving (or reacting) spheres^ namely, that in eclipses 
at perigee the sun is seen to be not wholly obscured, but to overlap 
with the edges of its circumference the circle of the moon, and to 
give light without hindrance. For if we accept this statement, 
then either the sun will show variation in its apparent diameter, 
or the moon will not, at its apogee, have its apparent diameter, 
as ascertained by observation, the same as that of the sun.' 
Cleomedes, too, alludes to the views of some of the more ancient 
astronomers who held that in total eclipses of the sun a bright rim 

1 Aristotle, Metaph. Λ. 8, 1073 b 10-17. 

2 Proclus, Hypotyposis astronomicarum positionum, c. 4, §§ 98, 99, p. 130, 
16-26, ed. Manitius, 



224 THEORY OF CONCENTRIC SPHERES 

of the sun was visible all round (Cleomedes' words would imply- 
that they asserted this to be true for all total eclipses, which is 
presumably a misapprehension), but adds that the statement has 
not been verified by observation. 1 Schiaparelli infers that Sosigenes 
was aware of the variations of the apparent diameter of the sun, 
as well as of the moon, and thinks that his object in alluding 
to annular eclipses in the above passage quoted from Simplicius, 
where the subject is again that of revolving spheres, was to use 
as an argument against that theory the fact that the distance 
of the sun from us is variable. 

1 Cleomedes, De motu circulari ii. 4, p. 190, 19-26, Ziegler. 



XVII 

ARISTOTLE (continued) 

It was convenient to give Aristotle's modified system of concen- 
tric spheres in close connexion with the systems of Eudoxus and 
Callippus, and to reserve the rest of his astronomy for separate 
treatment. While his modification of the beautiful theory of 
Eudoxus and Callippus was far from being an improvement, 
Aristotle rendered real services to astronomy in other respects. 
Those services consisted largely of thoughtful criticisms, generally 
destructive, of opinions held by earlier astronomers, but Aristotle 
also made positive contributions to the science which are of sufficient 
value to make it impossible to omit him from a history of Greek 
astronomy. 

We have seen that he modified the purely geometrical hypotheses 
of Eudoxus and Callippus in a mechanical sense. A purely 
geometrical theory did not satisfy him ; he must needs seek to 
assign causes for the motions of the several concentric spheres. We 
may therefore conveniently begin this chapter with an account of 
his views on Motion. Motion, according to Aristotle, is, like Form 1 
and Matter, 2 eternal and indestructible, without beginning or end. 3 
Motion presupposes 2iprimum movens which is itself unmoved ; 4 for 
that which is moved, being itself subject to change, cannot impart 
an unbroken and uniform movement; 5 the pritnum movens, then, 
must be one, 6 unchangeable, absolutely necessary ; 7 there is nothing 
merely potential about it, no unrealized possibility; 8 it must there- 
fore be incorporeal, 9 indivisible, 10 and unconditioned by space, 11 as 

1 Metaph. Z. 8, 1033 b 16, Z. 9, 1034 b 7, Λ. 3, 1069 b 35, &c. 

2 Phys. i. 9, 192 a 22-32. 3 Metaph. Λ. 6, 1071 b 7. 

4 Ibid. 107 1 b 4. 5 Phys. viii. 6, 259 b 22 : c. 10, 267 a 24. 

6 Metaph. Λ. 8, 1073 a 25, 1074 a 36, &c. 7 Ibid. Λ. 7, 1072 b 7-1 1. 

8 Ibid. Λ. 6, 1071 b 12. 9 Ibid. Λ. 6, 1071 b 20. 

10 Ibid. Λ. 9, 1075 a 7. 

11 De caelo i. 9, 279 a 18 sq. ; Phys. viii. 10, 267 b 18. 

1410 Ο 



226 ARISTOTLE part ι 

well as motionless and passionless -, 1 it is absolute Reality and pure 
Energy, 2 that is, God. 3 In another aspect the primum movens is the 
Final Cause, pure Being, absolute Form, the object of thought and 
desire ; 4 God is Thought, 5 self-sufficient, 6 contemplating unceasingly 
nothing but itself, 7 the absolute activity of Thought, constituting 
absolute reality and vitality and the source of all life. 8 The primum 
movens causes all the movements in the universe, not by any activity 
of its own 9 — for that would be a movement and, as immaterial, it 
can have no share in movement — but by reason of the fact that all 
things strive after it and try to realize, so far as possible, its Form ; 10 
it operates like a beloved object, and that which is moved by it 
communicates its motion to the rest. 11 

Motion takes place only by means of continuous contact between 
the motive principle and the thing moved. Aristotle insists upon 
this even in a case where the contact might seem to be only 
momentary, e. g. where a thing is thrown. The motion in that case 
seems to continue after contact with the thrower has ceased, but 
Aristotle will not admit this ; he assumes that the thrower moves 
not only the thing thrown but also the medium through which the 
thing is thrown, and makes the medium able to act as moved and 
movent at the same time (i.e. to communicate the movement); and 
further that the medium can continue to be movent even after it 
has ceased to be moved. 12 God then, as the first cause of motion, 
must be in contact with the world, 13 though Aristotle endeavours to 
exclude contiguity in space from the idea of ' contact ', which he 
often uses in the sense of immediate connexion, as of thought with 
its object. 14 The primum movens operates on the universe from the 
circumference, because the quickest motion is that of the (outermost 
limit of the) universe, and things move the quickest which are 
nearest to that which moves them. 15 Hence in a sense it could be 



1 De anima iii. 2, 426 a 10. 2 Metaph. Λ. 7, 1072 a 25. s De caelo, loc. cit. 
Metaph. Λ. 7, 1072 a 26 ; De anima iii. 10, 433 a 18. 
Eth. A r . x. 8, 1178 b 21 ; Metaph. Λ. 9, 1074b 25. 
De caelo ii. 12, 292 b 5 ; Politics, H. 1, 1323 b 23. 
Metaph. Λ. 9, 1075 a 10. 8 Metaph. Λ. 7, 1072 b 28. 

De caelo ii. 12, 292 a 22 ; Eth. N. x. 8, 1 178 b 20. 



10 Metaph. Λ. 7, 1072 a 26. n Ibid. 1072 b 3. 

12 Phys. viii. 10, 266 b 27 - 267 a 18. 13 De gen. et corr. i. 6, 323 a 31. 

14 

15 



Metaph. Θ. 10, 1051 b 24 ; Λ. 7, 1072 b 21. 
Phys. viii. 10, 267 b 7-9. 



ch. χνπ ARISTOTLE 227 

said that God is to Aristotle 'the extremity of the heaven'; 1 but 
Aristotle is careful to deny that there can be any body or space or 
void outside the universe ; what is outside is not in space at all ; the 
' end of the whole heaven ' is life (αιών), immortal and divine \ 2 

Motion in space is of three kinds, motion in a circle, motion in 
a straight line, and motion compounded of the two (' mixed '). 3 
Which of these can be endless and continuous ? The ' mixed ' 
would only be so if both the two components could ; but move- 
ment in a straight line cannot have this character, since every finite 
rectilinear movement has terminal points at which it must turn 
back, 4 and an infinite rectilinear movement is impossible, both in 
itself, 5 and because the universe is finite ; hence circular motion is 
the only motion which can be without beginning or end. 6 Simple 
bodies have simple motions ; thus the four elements tend to move 
in straight lines ; earth tends downwards, fire upwards ; between 
the two are water, the relatively heavy, and air, the relatively light. 
Thus the order, beginning from the centre, in the sublunary sphere 
is earth, water, air, fire. 7 Now, says Aristotle, simple circular 
motion is more perfect than motion in a straight line. As, then, 
there are four elements to which rectilinear motion is natural and 
circular motion not natural, so there must be another element, 
different from the four, to which circular motion is natural. 8 This 
element is superior to the others in proportion to the greater perfec- 
tion of circular motion and to its greater distance from us; 9 circular 
motion admits no such contraries as c up ' and ' down ' ; the superior 
element therefore can neither be heavy nor light; 10 the same absence 
of contrariety suggests that it is without beginning or end, im- 
perishable, incapable of increase or change (because all becoming 
involves opposites and opposite motions). 11 This superior element 
which fills the uppermost space is called ' aether ', 12 the * first ele- 

1 Sextus Emp. Adv. Math. x. 33 ; Hypotyp. iii. 218. 

2 De caelo i. 9, 279 a 16-28. 3 Phys. viii. 8, 261 b 29. 

4 Phys. viii. 8, 261 b 31-4. 5 Ibid. iii. 5, 206 a 7 ; c. 6, 206 a 16. 

6 Ibid. viii. 8, 261 a 27 - 263 a 3, 264 a 7 sqq. ; c. 9, 265 a 13 sq. 

7 Aristotle is careful, however, to explain that the division between air and fire 
is not a strict one, as between two definite layers ; there is some intermixture 
(cf. Meteor, i. 3, 341 a 1-9). Further, the ί fire ' is what from force of habit we 
catl fire ; it is not really fire, for fire is an excess of heat, a sort of ebullition 
(ibid. 340 b 22, 23). 8 De caelo i. 2, 268 b 26 - 269 b 17. 9 Ibid. 

10 De caelo i. 3, 269 b 18-33. u Ibid. 270 a 12-35. 

12 Ibid. 270 b 1-24. 

Q2 



228 ARISTOTLE parti 

ment ', 1 or ' a body other and more divine than the four so-called 
elements'; 2 its changelessness is confirmed by long tradition, which 
contains no record of any alteration in the outer heaven itself or in 
any of its proper parts. 3 Of this element are formed the stars, 4 
which are spherical, 5 eternal, 6 intelligent, divine. 7 It occupies the 
whole region from the outside limit of the universe down to the 
orbit of the moon, though it is not everywhere of uniform purity, 
showing the greatest difference where it touches the sublunary 
sphere. 8 Below the moon is the terrestrial region, the home of the 
four elements, which is subject to continual change through 
the strife of those elements and their incessant mutual transfor- 
mations. 9 

There is, Aristotle maintains, only one universe or heaven, and 
that universe is complete, containing within it all the matter there 
is. For, he argues, all the simple bodies move to their proper 
places, earth to the centre, aether to the outermost region of the 
universe, and the other elements to the intervening spaces. There 
can be no simple body outside the universe, for that body has its 
own natural place inside, and, if it were kept outside by force \ the 
place occupied by it would be the natural place for some other 
body; which is impossible, since all the simple bodies have their 
proper places inside. The same argument holds for mixed bodies ; 
for, where mixed bodies are, there also are the simple bodies of 
which they are composed. Nor can there be any space or void out- 
side the universe, for space or void is only that in which a body is 
or can be. 10 Another argument is that the primum movens is single 
and complete in itself; hence the world, which derives its eternal 
motion from the primum movens, must be so too. 11 If it be sug- 
gested that there may be many particular worlds as manifestations 
of one concept ' world ', Aristotle replies that this cannot be ; for 
the heaven is perceptible to our senses ; hence it and other heavens 



1 De caelo iii. i, 298 b 6 ; Meteor, i. 1, 338 b 21, &c. 

2 De gen. an. ii. 3, 736 b 29-31. 3 De caelo i. 3, 270b 11-16. 

4 De gen. an. ii. 3, y^7 a Ι• 5 De caelo ii. 8, 290 a 7-b II. 

β Metaph. Λ. 8, 1073 a 34. 

7 Ibid. 1074a 38-b 3 ; Eth. N. vi. 1 143 b 1. 8 Meteor, i. 3, 340 b 6-10. 

9 Meteor, ii. 3, 357 b 30. 
10 De caelo i. 9, 278 b 8 - 279 a 14. 
' Metaph. A. 8, 1074 a 36-8. 



11 



ch. χνπ ARISTOTLE 229 

(if any) must contain matter ; but our heaven contains all the matter 
there is, and therefore there cannot be any other. 1 

Next, the universe is finite. In the Physics Aristotle argues that 
an infinite body is inconceivable, thus. An infinite body must 
either be simple or composite. If composite, it is composed of 
elements ; these are limited in number ; hence an infinite body 
could only be made up of them if one or more were infinite in 
magnitude ; but this is impossible, because there would then be no 
room for the rest. Neither can it be simple ; for no perceptible 
simple body exists except the elements, and it has been shown that 
none of them can be infinite. 2 In the De caelo he approaches the 
subject from the point of view of motion. A body which has 
a circular motion, as the universe has, must be finite. For, if it is 
infinite, the straight line from the centre to a point on its circum- 
ference must be infinite ; now if, as being infinite, this distance can 
never be traversed, it cannot revolve in a circle, whereas we see 
that in fact the universe does so revolve. 3 Further, in an infinite 
body there can be no centre ; hence the universe which rotates about 
its centre cannot be infinite. 4 

Aristotle's arguments for the spherical shape of the universe are 
of the usual kind. As the circle, enclosed by one line, is the first of 
plane figures, so the sphere, bounded by one surface, is the first 
of solid figures ; hence the spherical shape is appropriate to the 
1 first body', the subject of the Outermost revolution \ 5 Next, as 
there is no space or void outside the universe, it must, as it revolves, 
continually occupy the same space ; therefore it must be a sphere ; 
for, if it had any other form, this condition would not be satisfied. 6 
[Aristotle is not strictly correct here, since any solid of revolution 
revolving round its axis always occupies the same space, but it is 
true that only a sphere can remain in exactly the same position when 
revolving about any diameter whatever.] Further, we may infer the 
spherical form of the universe from the bodies in the centre. We 
have first the earth, then the water round the earth, air round the 
water, fire round the air, and similarly the bodies above the fire ; 

1 De caelo i. 9, 277 b 27 - 278 a 28. 

2 Phys. iii. 5, 204 b 3-35. 3 De caelo i. 5, 271 b 28 - 272 a 7. 
4 De caelo i. 7, 275 b 12-15. 5 Ibid. ii. 4, 286 b 10-287 a 5. 

6 Ibid. ii. 4, 287 a 11-22. 



23° 



ARISTOTLE part ι 



now the surface of the water is spherical ; hence the surfaces of the 
layers following it, and finally the outermost surface, correspond. 1 

The fabric of the heavens is made up of spherical shells, as it 
were, one packed inside the other so closely that there is no void 
or empty space between them ; 2 this applies not only to the astral 
spheres, 3 but right down to the earth in the middle ; 4 it is necessary 
so far as the moving spheres are concerned because there must 
always be contact between the moving and the moved. 5 

We have above described the working of Aristotle's mechanical 
system of concentric spheres carrying the fixed stars and producing 
the motions of the planets respectively, and it only remains to add 
a word with reference to the motive power acting on the spheres 
other than that of the 'outermost revolution'. The outermost 
sphere, that of the fixed stars, is directly moved by the one single 
and eternal primum movens, Divine Thought or Spirit. Only one 
kind of motion is produced when one movens acts on one object ; 6 
how then do we get so many different movements in the spheres 
other than the outermost? Aristotle asks himself this question : 
Must we suppose that there is only one unmoved movens of the 
kind, or several, and, if several, how many are there ? He is obliged 
to reply that, as eternal motion must be due to an eternal move?is, 
and one such motion to one such movens, while we see that, in addi- 
tion to the simple revolution of the whole universe caused by the 
unmoved primum movens^ there exist other eternal movements, 
those of the planets, we must assume that each of the latter move- 
ments is due to a substance or essence unmoved in itself and 
eternal, without extension in space. 7 The number of them must be 
that of the separate spheres causing the motion of the separate 
planets. 8 The number of these spheres he had, as we have seen 
(p. 217), fixed provisionally, while recognizing that the progress of 
astronomy might make it necessary to alter the figures. 9 Of the 
several spheres which act on any one planet, the first or outermost 
alone is moved by its own motion exclusively ; each of the inner 
spheres, besides having its own independent movement, is also 

1 De caelo ii. 4, 287 a 30-b 4. 

2 Cf. Phys. vii. 2, 243 as. 3 De caelo i. 9, 278 b 16-18. 

4 De caelo ii. 4, 287 a 5-1 1. 5 Phys. vii. 1, 242 b 24-6, vii. 2, 243 a 3-5. 

6 Phys. viii. 6, 259 a 18. 7 Metaph. Λ. 8, 1073 a 14-b 10. 

8 Metaph. Λ. 8. 1074 a 13-16. 9 Ibid. 1073 b 10-17. 



ch.xvii ARISTOTLE 231 

carried round in the motion of the next sphere enveloping it, so that 
all the inner spheres, while themselves movent, are also moved by 
the eternal unmoved movent. 1 

In a chapter of the De caelo 2, Aristotle discusses the question 
which is the right side of the heaven and which the left. The dis- 
quisition is not important, but it is not unamusing. 3 He begins with 
a reference to the view of the Pythagoreans that there is a right 
and a left in the universe, and proceeds to investigate whether the 
particular distinction which they draw is correct or not, ■ assuming 
that it is necessary to apply such principles as " right " and " left" 
to the body of the universe \ 4 There being three pairs of such 
opposites, up and down (or upper and lower), right and left, before 
and behind (or forward and backward), he begins with the distinc- 
tions (1) that * up ' is the principle of length, ' right' of breadth, and 
ζ before ' of depth, and (2) that ' up ' is the source of motion {o6ev ή 
κίνησις), 'right' the place from which it starts (αφ' ov) i and ' to the 
front ' (e/y το πρόσθζν) is the place to which it is directed (ίψ 0). 
Now the fact that the shape of the universe is spherical, alike in all 
its parts, and continually in motion, is no obstacle to calling one 
part of it ' right ' and the other ' left '. What we have to do is to 
think of something which has a right and left of its own (say a man) 
and then place a sphere round it. 5 Now, says Aristotle, I call the 
diameter through the two poles the length of the universe (because 
only the poles remain fixed), so that I must call one of the poles 
the upper, and the other the lower. He then proceeds to show 
that the proper relativities can only be preserved by calling the 
south (the invisible) pole the upper and the north (the visible) 
pole the lower, from which it follows that we live in the lower and 
left hemisphere, and the inhabitants of the regions towards the south 
pole live in the upper and right hemisphere ; and this is precisely 
the opposite of what the Pythagoreans hold, namely that we live 
in the upper and right hemisphere, and the antipodes represent 
the lower and left. The argument amounts to this. ' Right ' is the 
place from which motion in space starts ; and the motion of 

1 Phys. viii. 6, 259 b 29-31. 2 De caelo ii. 2, 284 b 6 - 286 a 2. 

3 This matter also is fully discussed by Boeckh, Das kosmische System des 
Platon, pp. 1 12-19. 
* De caelo ii. 2, 284 b 9-10. 5 Ibid. 285 b 1-3. 



232 ARISTOTLE part ι 

the heaven starts from the side where the stars rise, i.e. the east ; 
therefore the east is ' right' and the west is 'left'. If now (i) you 
suppose yourself to be lying along the world's axis with your head 
towards the north pole, your feet towards the south pole, and your 
right hand towards the east, then clearly the apparent motion of the 
stars from east to west is over your back from your right side towards 
your left ; this motion, Aristotle maintains, cannot be called motion 
' to the right ', and therefore our hypothesis does not fit the assump- 
tion from which we start, namely that the daily rotation ' begins 
from the right and is carried round towards the right (επί τα δεξιά) ', 1 
We must therefore alter the hypothesis and suppose (2) that you 
are lying with your head towards the south pole and your feet 
towards the north pole. If then your right hand is to the east, the 
daily motion begins at your right hand and proceeds over the front 
of your body from your right hand to your left. We should nowa- 
days regard this as giving precisely the wrong result, since motion 
round us in front from right to left can hardly be described as επί 
τα δεξιά, ' to the right'; so that hypothesis (1) would, to us, seem 
preferable to hypothesis (2). But Aristotle's point of view is fairly 
clear. We are to suppose a man (say) standing upright and giving 
a horizontal turn with his right hand to a circle about a vertical 
diameter coincident with the longitudinal axis of his body. Aris- 
totle regards him as turning the circle towards the right when he 
brings his right hand towards the front of his body, although we 
should regard it as more natural to apply ' towards the right ' to a 
movement of his right hand still more to the right, i.e. round by 
the right to the back. The (to us) unnatural use of the terms 
by Aristotle is attested by Simplicius who says that motion επί 
δεξιά is in any case towards the front (πάντως eh το έμπροσθεν 
εστί), 2 and it is doubtless due to what Aristotle would regard as the 
necessity of making front (in the dichotomy front and back, or 
before and behind) correspond to right (in the dichotomy right and 
left), just as up (in the dichotomy up and down) must also corre- 
• spond to right ; this is indeed clear from his own statement quoted 
above that, as 'the right' is the place from which motion starts, so 
' to the front ' is the place towards which it is directed. 

We come next to Aristotle's view as to the shape of the heavenly 
1 De caelo ii. 2, 285 b 20. 2 Simplicius on De caelo, p. 392, 1, Heib. 



CH. χνπ ARISTOTLE 233 

bodies and the arguments by which he satisfied himself that they 
do not move of themselves but are carried by material spheres. 

He held that the stars are spherical in form. One argument in 
support of this contention is curious. Nature, he says, does nothing 
without a purpose ; Nature therefore gave the stars the shape 
most unfavourable for any movement on their own part ; she denied 
to them all organs of locomotion, nay, made them as different as 
possible from the things which possess such organs. With this end 
in view, Nature properly made the stars spherical ; for, while the 
spherical shape is the best adapted for motion in the same place 
(rotation), it is the most useless for progressive motion. 1 This is 
in curious contrast to the view of Plato who, with more reason, 
regarded the cube as being the shape least adapted for motion 
(άκινητότατον)? The second argument is from analogy. Since 
the moon is shown by the phases to be spherical, while we see 
similar curvature in the lines separating the bright part of the sun 
from the dark in non-total solar eclipses, we may conclude from this 
that the sun and, by analogy, the stars also are spherical in form. 3 

With regard to the spheres carrying the stars round with them, 
we note first that the ' heaven ', in the sense of the ' outermost 
heaven ' or ' the outermost revolution of the All ' (77 έσχατη περι- 
φορά, του παντός), which is the sphere of the fixed stars, was with 
Aristotle a material thing, a ' physical body ' (σώμα φυσικον).* 
Now, says Aristotle, 5 seeing that both the stars and the whole 
heaven appear to change their positions, there are various a priori 
possibilities to be considered ; (1) both the stars and the heaven 
may be at rest, (2) both the stars and the heaven may be in motion, 
or (3) the stars may move and the heaven be at rest, or vice versa. 
Hypothesis (1) is at once ruled out because, under it, the 
observed phenomena could not take place consistently with 
the earth being at rest also ; and Aristotle assumes that the earth 
is at rest (την Se γήν ύποκβίσθω ήρεμςΐν). Coming to hypothesis (2), 
we have to remember that the effect of a uniform rotation of the 
heaven about an axis passing through the poles is to make par- 
ticular points on this spherical shell describe parallel circles about 

1 De caelo ii. 8, 29oa3i-bs ; c. II, 291b 11-17. 

2 Plato, Timaeus 55 D, E. 3 De caelo ii. 11, 291 b 17-23. 
4 De caelo i. 9, 278 b 11-14. 5 Ibid. ii. 8, 289 b 1 sqq. 



234 ARISTOTLE parti 

the axis ; suppose then that the heaven rotates in this way, and that 
the stars also move. Now, says Aristotle, the stars and the circles 
cannot move independently ; if they did, it is inconceivable that the 
speeds of the stars would always be exactly the same as the speeds 
of the circles ; for, while the speeds of the circles must necessarily 
be in proportion to their sizes, i.e. to their radii, it is not reasonable 
to suppose that the stars, if they moved freely, would revolve at 
speeds proportional to the radii of the circles ; yet they would have 
to do so if the stars and the circles are always to return to the same 
places at the same times, as they appear to do. Nor can we, as 
in hypothesis (3), suppose the stars to move and the heaven to be 
at rest ; for, if the heaven were at rest, the stars would have to move 
of themselves at speeds proportional to the radii of the circles they 
describe, which has already been stated to be an unreasonable sup- 
position. Consequently only one possibility remains, namely that 
the circles alone move, and the stars are fixed on them and carried 
round with them ;* that is, they are fixed on, and carried round with, 
the sphere of which the circles are parallel sections. 

Again, says Aristotle, there are other considerations which sug- 
gest the same conclusion. If the stars have a motion of their own, 
they can, being spherical in shape, have only one of two movements, 
namely either (1) whirling (δίνησις) or (2) rolling (kvXlvis). Now 
(1), if the stars merely whirled or rotated, they would always 
remain in the same place, and would not move from one position 
to another, as everybody admits that they do. Besides, if one 
heavenly body rotated, it would be reasonable to suppose that they 
all would. But, in fact, the only body which seems to rotate is the 
sun and that only at the times of its rising and setting ; this, how- 
ever, is only an optical illusion due to the distance, * for our sight, 
when at long range, wavers' (literally 'turns' or 'spins', έλίσσ-ςται). 
This, Aristotle incidentally observes, may perhaps be the reason 
why the fixed stars, which are so distant, twinkle, while the planets, 
being nearer, do not. It is thus the tremor or wavering of our 
sight which makes the heavenly bodies seem to rotate. 2 In thus 
asserting that the stars do not rotate^ Aristotle is of course opposed 
to Plato, who held that they do. 3 

1 De caelo ii. 8, 289532. 2 Ibid. ii. 8, 290 a 9-23. 3 Plato, Timaeus 40 a. 



ch.xvii ARISTOTLE 235 

Again (2), if the stars rolled (along, like a wheel), they would 
necessarily turn round ; but that they do not turn round in this 
way is proved by the case of the moon, which always shows us one 
side, the so-called face. 1 

It is for a particular reason that I have reproduced so fully 
Aristotle's remarks about rotation and rolling as conceivable move- 
ments for stars as spherical bodies. It has been commonly re- 
marked that Aristotle draws a curious inference from the fact that 
the moon has one side always turned to us, namely that the moon 
does not rotate about its own axis, whereas the inference should be 
the very opposite. 2 But this is, I think, a somewhat misleading 
statement of the case and less than just to Aristotle. What he says 
is that the moon does not turn round in the sense of rolling along ; 
and this is clear enough because, if it rolled along a certain path, it 
would roll once round while describing a length equal to 3-1416 
times its diameter, but it manifestly does not do this. But Aris- 
totle does not say that the moon does not rotate ; he does not, it is 
true, say that it does rotate either, but his hypothesis that it is 
fixed in a sphere concentric with the earth has the effect of keeping 
one side of the moon always turned towards us, and therefore inci- 
dentally giving it a rotation in the proper period, namely that of its 
revolution round the earth. I cannot but think that the fact of the 
moon always showing us one side was one of the considerations, if 
not the main consideration, which suggested to Aristotle that the stars 
were really fixed in material spheres concentric with the earth. 

We pass to matters which are astronomically more important. 
And first as to the spherical shape of the earth. Aristotle begins 
by answering an objection raised by the partisans of a flat earth, 
namely that the line in which the horizon appears to cut the sun as 
it is rising or setting is straight and not curved. 3 His answer is 
confused ; he says that the objectors do not take account of the 
distance of the sun from the earth and of the size of its circum- 
ference, 4 the fact being that you can have an apparently straight line 

1 De caelo ii. 8, 290 a 26. 

% Cf. Martin, ' Hypotheses astronomiques grecques ' in Memoires de Γ Acad, 
des Inscriptions et Belles-Lettres, xxx. 1881, p. 287 ; Dreyer, Planetary Systems, 
p. in, note. 8 De caelo ii. 13, 294a 1. 

4 της περιφερύαε seems clearly to be the circumference of the sun (not that of 
the horizon which cuts the solar disc). 



236 ARISTOTLE part ι 

as a section when you see it from afar in a circle which on account 
of its distance appears small. He should no doubt have said, first, 
that the sun, as we see it, looks like a flat disc of small size on 
account of its distance, and then that the section of an object 
apparently so small by the horizon is indistinguishable from a 
section by a plane through our eye, so that the section of the disc 
appears to be a straight line. He has, however, some positive 
proofs based on observation, (i) In partial eclipses of the moon 
the line separating the bright from the dark portion is always 
convex (circular) — unlike the line of demarcation in the phases of 
the moon, which may be straight or curved in either direction — 
this proves that the earth, to the interposition of which lunar 
eclipses are due, must be spherical. 1 He should no doubt have 
said that a sphere is the only figure which can cast a shadow such 
that a right section of it is always a circle ; but his explanation 
shows that he had sufficiently grasped this truth. (2) Certain stars 
seen above the horizon in Egypt and in Cyprus are not visible 
further north, and, on the other hand, certain stars set there which 
in more northern latitudes remain always above the horizon. As 
there is so perceptible a change of horizon between places so near 
to each other, it follows not only that the earth is spherical, but 
also that it is not a very large sphere. He adds that this makes it 
not improbable that people are right when they say that the region 
about the Pillars of Heracles is joined on to India, one sea connect- 
ing them. It is here, too, that he quotes the result arrived at 
by mathematicians of his time, that the circumference of the earth 
is 400,000 stades. 2 He is clear that the earth is much smaller 
than some of the stars. 3 On the other hand, the moon is smaller 
than the earth. 4 Naturally, Aristotle has a priori reasons for the 
sphericity of the earth. Thus, using once more his theory of heavy 
bodies tending to the centre, he assumes that, whether the heavy 
particles forming the earth are supposed to come together from all 
directions alike and collect in the centre or not, they will arrange 
themselves uniformly all round, i.e. in the shape of a sphere, since, if 
there is any greater mass at one part than at another, the greater 



1 De caelo ii. 14, 297 b 23-30. 2 Ibid. 297 b 30 - 298 a 20. 

3 Ibid. 298 a 19 ; Meteor, i. 3, 339 b 7-9. 

4 Aetius, ii. 26. 3 (D. G. p. 357 b 11). 



ch.xvii ARISTOTLE 237 

mass will push the smaller until the even collection of matter all 
round the centre produces equilibrium. 1 

Aristotle's attempted proof that the earth is in the centre of the 
universe is of course a petitio principii. He begins by attempting 
to refute the Pythagorean theory that the earth, like the planets and 
the sun and moon, moves round the central fire. The Pythagoreans, 
he says, conceived the central fire to be the abode of sovereignty in 
the universe, the Watchtower of Zeus, while others might say that 
the centre, being the worthiest place, is appropriate for the worthiest 
occupant, and that fire is worthier than earth. To this he replies 
that the centre of a thing is not so worthy as the extremity, for it 
is the extremity which limits or defines a thing, while the centre is 
that which is limited and defined, and is more like a termination 
than a beginning or principle. 2 When Aristotle comes to state his 
own view, he rightly says that heavy bodies, e.g. parts of the earth 
itself, tend towards the centre of the earth ; for bodies which fall 
towards the earth from different places do not fall in parallel lines 
but ' at equal angles ', i.e. at right angles, to the (spherical) surface 
of the earth, and this proves that they fall in the direction of its 
centre. Similarly, if a weight is thrown upwards, however great the 
force exerted, it falls back again towards the centre of the earth. 3 But, 
he asks, do bodies tend towards the centre because it is the centre 
of the universe or because it is the centre of the earth, ' since both 
have the same centre ' ? 4 He replies that they must tend towards 
the centre of the universe because, in the reverse case of the light 
elements, e.g. fire, it is the extremities of the space which envelops 
the centre (i.e. the extremities of the universe) to which they 
naturally tend. 5 Even the show of argument in the last sentence 
does not prevent the whole of the reasoning from being a petitio 
principii. For it is exclusively based on the original assumption 
that, of the four elements, earth and, next to that, water tend to 
move in a straight line 'downwards', i.e., on Aristotle's view, 
towards the centre of the universe, 6 the effect of which is that not 

1 De caelo ii. 14, 297 a 8-b 18. 

2 Ibid. ii. 13, 293ai7-bi5• 3 Ibid. ii. 14, 296b 18-25. 
4 Ibid. 296 b 9-12. 5 Ibid. 296 b 12-15. 

6 Phys. iv. 4, 212 a 26 ; c. 8, 214 b 14 ; and especially De caelo iv. 1. 308 a 15- 
31. In the last-cited passage Aristotle, without mentioning Plato by name, 
attacks Plato's doctrine that, in a perfect sphere such as the universe is, you 



238 ARISTOTLE part 1 

only do the particles of the earth tend to the centre of the universe, 
but a fortiori the earth itself, which must therefore occupy the 
centre of the universe. 1 

Another argument is that, according to the astronomical views of 
the mathematicians, the phenomena which are observed as the 
heavenly bodies change their positions relatively to one another are 
just what they should be on the assumption that the earth is in the 
centre. 2 The answer to this is, as Martin says, ' How do you 
know ? And how can you use the argument when you have quoted, 
without stating any objection to it, the argument of the Pytha- 
goreans that their theory of the motion of the earth need cause no 
sensible difference of parallax in comparison with the theory that 
the earth is at the centre ? ' 

The earth being in the centre of the universe, what keeps it 
there? Dealing with this question, Aristotle again begins by a 
consideration of the views of earlier philosophers. He rejects 
Thales' view that the earth floats on water as contrary to experi- 
ence, since earth is heavier than water, and we see water resting or 
riding on earth, but not the reverse. He rejects, too, the view of 
Anaximenes, Anaxagoras, and Democritus that it rides on the air 
because it is flat and, acting like a lid to the air below it, is sup- 
ported by it. Aristotle points out first that, if it should turn out 
that the earth is round and not flat, it cannot be the flatness which 
is the reason of the air supporting it ; according to the argument it 

cannot properly describe one part rather than another as ' up ' or ' down ' ; on the 
contrary, says Aristotle, I call the centre, where heavy bodies collect, ' down ', 
and the extremities of the sphere, whither light bodies tend to rise, ' up '. But, 
as usual, there is less difference between the two views than Aristotle would 
have us believe. Plato {Timaeus 62 D) said, it is true, that, as all points of the 
circumference are equidistant from the centre, it is incorrect to apply the terms 
' up ' and ' down ' to different specific portions of that circumference, or to 
call any portion of the sphere ' up ' or ' down ' relatively to the centre, which 
is neither ' up ' nor ' down ', but simply the centre. But he goes on to say 
(63 b-e) that you can use the terms in a purely relative sense ; any two 
localities may be * up ' and ' down ' relatively to one another, and Plato proposes 
a criterion. If a body tends to move to a certain place by virtue of seeking for 
its like, this tendency is what constitutes its heaviness, and the place to which 
it tends is ' down ' ; and the opposite terms have the opposite meanings. The 
only difference made by Aristotle is in definitely allocating the centre of the 
universe as the place of the heaviest element, earth, and arranging the other 
elements in order of lightness in spherical layers round it, so that on his system 
the centre of the universe becomes ' down ', and any direction outwards along 
a radius is 'up'. 

1 De caelo ii. 14, 296 b 6-9. 2 Ibid. 297 a 2-6. 



ch. χνπ ARISTOTLE 239 

must rather be its size than its flatness, and, if it were large enough, 
it might even be a sphere. With this he passes on. 1 Nor does 
Empedocles' theory meet with more favour ; if the earth is kept 
in its place in the same way as water in a cup which is whirled 
round, this means that the earth is kept in its place by force ; and to 
this view Aristotle opposes his own theory that the earth must 
have some natural tendency, and a proper place, of its own. Even 
assuming that it came together by the whirling of a vortex, why do 
all heavy bodies now tend towards it ? The whirling is at all events 
too far away from us to cause this. And why does fire move 
upwards ? This cannot be through the whirling either ; and, if fire 
naturally tends to move to a certain region, surely the earth should 
too. But, indeed, the heavy and the light were prior to the whirling, 
and what determines their place is, not whirling, but the difference 
between ' up ' and ' down '. Finally he deals with the view of 
Anaximander (followed by Plato) that the earth is in equilibrium 
through being equidistant from all points of the circumference, and 
therefore having no reason to move in one direction rather than 
another. Incidentally comparing the arguments (1) that, if you 
pull a hair with force and tension exactly equal throughout, it will 
not break, and (2) that a man would have to starve if he had 
victuals and drink equally disposed all round him, Aristotle again 
complains that the theory does not take account of the natural 
tendency of one thing to move to the centre, and of another to 
move to the circumference. It happens incidentally to be true that 
a body must remain at the centre if it is not more proper for it to 
move this way or that, but whether this is so or not depends on the 
body ; it is not the equidistance from the extremities which keeps 
it there, for the argument would require that, if fire were placed in 
the centre, it would remain there, whereas in fact it would not, 
since its tendency to fly upwards would carry it uniformly in all 
directions towards the extremities of the universe ; hence it is not 
the equidistance, but the natural tendency of the body, which 
determines the place where it will rest. 3 

In setting himself to prove that the earth has no motion what- 
ever, Aristotle distinguishes clearly between the two views (1) of 

1 De caelo ii. 13, 294 a 28-b 30. 2 Ibid. 295 a 16-b 9. 

3 Ibid. 295 b 10 - 296 a 21. 



24o ARISTOTLE parti 

those who give it a motion of translation or ' make it one of the 
stars', and (2) of those who regard it as packed and moving about an 
axis through its centre. 1 Though he arbitrarily adds the words ' and 
moves ' (και κινβΐσθαι) to the phraseology of the Timaeus, thereby 
making it appear that Plato attributed to the earth a rotation about 
its axis, which, as we have seen, he could not have done, the second 
of the two views was actually held by Heraclides Ponticus, who 
was Aristotle's contemporary. It seems likely, as Dreyer suggests, 2 
that, in speaking of a motion of the earth 'at the centre itself', 3 
Aristotle is not thinking of a rotation of the earth in twenty-four 
/tours, i.e. a rotation replacing the apparent revolution of the fixed 
stars, as Heraclides assumed that it did ; for he does not mention 
the latter feature or give any arguments against it ; on the con- 
trary, he only deals with the general notion of a rotation of the 
earth, and moreover mixes up his arguments against this with his 
arguments against a translation of the earth in space. He uses 
against both hypotheses his fixed principle that parts of the earth, 
and therefore the earth itself, move naturally towards the centre. 4 
Whether, he says, the earth moves away from the centre or at the 
centre, such movement could only be given to it by force ; it could 
not be a natural movement on the part of the earth because, if 
it were, the same movement would also be natural to all its parts, 
whereas we see them all tend to move in straight lines towards the 
centre ; the assumed movement, therefore, being due to force and 
against nature, could not be everlasting, as the structure of the 
universe requires. 5 

The second argument, too, though directed against both hypo- 
theses, really only fits the first, that of motion in an orbit. 

' Further, all things which move in a circle, except the first (outer- 
most) sphere, appear to be left behind and to have more than one 
movement ; hence the earth, too, whether it moves about the 
centre or in its position at the centre, must have two movements. 
Now, if this occurred, it would follow that the fixed stars would 
exhibit passings and turnings (παρόδους και τροπάς). This, how- 

1 De caelo ii. 13, 293 a 20-3, b 18-20 ; c. 14, 296 a 25-7. 

2 Dreyer, Planetary Syste?ns, pp. 116, 117. 

3 De caelo ii. 14, 296 a 29, b 2. 4 Ibid. 296 b 6-8. 
5 Ibid. 296 a 27-34. 



ch. χνπ ARISTOTLE 241 

ever, does not appear to be the case, but the same stars always rise 
and set at the same places on the earth.' 1 

The bodies which appear to be ' left behind and to have more 
movements than one* are of course the planets. The argument 
that, if the earth has one movement, it must have two, is based upon 
nothing more than analogy with the planets. Aristotle clearly 
inferred as a corollary that, if the earth has two motions, one must 
be oblique to the other, for it would be obliquity to the equator in 
at least one of the motions which would produce what he regards 
as the necessary consequence of his assumption, namely that the 
fixed stars would not always rise and set at the same places. As 
already stated, Aristotle can hardly have had clearly in his mind 
the possibility of one single rotation about the axis in tzventy-four 
hours replacing exactly the apparent daily rotation ; for he would 
have seen that this would satisfy his necessary condition that the 
fixed stars shall always rise and set at the same places, and there- 
fore that he would have to get some further support from elsewhere 
to his assumption that the earth must have two motions. Still 
less could he have dreamt of the possibility of Aristarchus's later 
hypothesis that the earth has an annual revolution as well as a 
daily rotation about its axis, which hypothesis satisfies, as a matter 
of fact, both the condition as to two motions and the condition as 
regards the fixed stars. 

The Meteorologica deals with the sublunary portion of the 
heavenly sphere, the home of the four elements and their combina- 
tions. Only a small portion of the work can be said to be astrono- 
mical, but some details bearing on our subject may be given. We 
have seen the four elements distinguished according to their relative 
heaviness or lightness, and the places which are proper to them 
respectively ; in the Meteorologica they are further distinguished 
according to the tangible qualities which are called their causes 
(αίτια). These tangible qualities are the two pairs of opposites, 
hot — cold, and dry — moist ; and when we take the four combinations 
of these in pairs which are possible we get the four elements ; hot 
and dry = fire, hot and moist = air (air being a sort of vapour), 
cold and moist = water, cold and dry = earth. 2 Of the four qualities 

1 De caelo ii. 14, 296 a 34-b 6 2 De gen. et corr. ii. 3, 330 a 30-b 7. 

1410 R 



242 ARISTOTLE part ι 

two, hot and cold, are regarded as active, and the other two, dry 
and moist, as passive. 1 Since each element thus contains an active 
as well as a passive quality, it follows that all act upon and are acted 
upon by one another, and that they mingle and are transformed into 
one another. 2 Every composite body contains all of them ; 3 they 
are never, in our experience, found in perfect purity. 4 Elemental fire 
is warm and dry evaporation, 5 not flame ; elemental fire is a sort of 
1 inflammable material ' which ' can often be kindled by even a little 
motion, like smoke'; 6 but flame, or fire in the sense of flame, is 
' an excess of heat or a sort of ebullition ', 7 or an ebullition of dry 
wind 8 or of dry heat 9 ; again, flame is said to be a fleeting, non- 
continuous product of the transformation of moist and dry in close 
contact. 10 The reason for this distinction between ' fire ' and flame 
is obvious, as Zeller says ; for Aristotle could not have made the 
outer portion of the terrestrial sphere, contiguous to the aether, to 
consist of actual burning flame. According to Aristotle, the stars 
are not made of fire (still less all the spaces between them); in 
themselves they are not even hot ; their light and heat come from 
friction with the air through which they move [notwithstanding 
that they are in the aethereal sphere] ; the air in fact becomes fire 
through their impact on it ; the stratum of air which lies nearest to 
them underneath the aethereal sphere is thus warmed. Especially 
is this the case with the sun ; the sun is able to produce heat in 
the place where we live because it is not so far off as the fixed 
stars and it moves swiftly (the stars, though they move swiftly, 
are far off, and the moon, though near to us, moves slowly) ; further, 
the motion often causes the fire surrounding the atmosphere to 
scatter and rush downwards. 11 

Such phenomena as shooting stars (διάττοντ€9 or διαθίοντες 
άστβρβς) and meteors (of the two kinds called δαλοί and cuyes) are 
next dealt with. These are due to two kinds of exhalation, one 
more vaporous (rising from the water on and in the earth), the other 



1 Meteor, iv. I, 378 b 10-13, 21-5. 

2 De gen. et corr. ii. 2, 329 b 22 sq. 3 Ibid. c. 8, 334 b 31 sq. 

4 Ibid. c. 3, 330 b 21 ; Meteor, ii. 4, 359 b 32, &c. 

5 Meteor, i. 3, 340 b 29 ; c. 4, 341 b 14. 6 Ibid. 341 b 19-21. 
7 Ibid. c. 3, 340 b 23. 8 Ibid. c. 4, 341 b 21-2. 

9 De gen. et corr. ii. 3, 330 b 29. 10 Meteor, ii. 2, 355 a 9. 

11 De caelo ii. 7, 289 a 13-35 '> Meteor, i. 3, 340 a 1, 341 a 12-36. 



ch. χνπ ARISTOTLE 243 

dry and smoke-like (rising from the earth) ; these go upwards, the 
latter uppermost, the former below it, until, caught in the rotation 
at the circumference of the sublunary sphere, they take fire. The 
particular varieties of appearance which they present depend on 
the shape of the rising exhalations and the inclination at which 
they rise. Sometimes, however, they are the result, not of motion 
kindling them, but of heat being squeezed out of air which comes 
together and is condensed through cold ; in this case their motion 
is like a throw (ρΐψις) rather than a burning, being comparable to 
kernels or pips of fruits pressed between our fingers and so made to 
fly to a distance ; this is what happens when the star falls down- 
wards, since but for such compelling force that which is hot would 
naturally always fly upwards. All these phenomena belong to the 
sublunary sphere. 1 The aurora is regarded as due to the same 
cause combined with reflection lighting up the air. 2 

Aristotle has two long chapters on comets. 3 He begins, as 
usual, by reviewing the opinions of earlier philosophers and so 
clearing the ground. Anaxagoras and Democritus had explained 
comets as a ' conjunction of the planets when, by reason of coming 
near, they seem to touch one another'. Some of 'the so-called 
Pythagoreans ' thought that they were one planet, which we 
only see at long intervals because it does not rise far above the 
horizon, the case being similar to that of Mercury, which, since it 
only rises a little above the horizon, makes many appearances 
which are invisible to us and is actually seen at long intervals only. 
Hippocrates of Chios and his pupil Aeschylus gave a similar 
explanation but added a theory about the tail. The tail, they said, 
does not come from the comet itself, but the comet, as it wanders 
through space, sometimes takes on a tail ' through our sight being 
reflected, at the sun, from the moisture attracted by the comet '. 
Explanations by Hippocrates and Aeschylus follow, of the reasons 
(1) of the long intervals between the appearances of a comet: the 
reason in this case being that it is only left behind by the sun very 
slowly indeed, so that for a long time it remains so close to the sun 
as not to be visible ; (2) of the impossibility of a tail appearing 
when the comet is between the tropical circles or still further 

1 Meteor, i. 4, 341 b-342 a. 2 Ibid. c. 5, 342 a 34-b 24. 

8 Ibid. cc. 6, 7, 342 b 25 - 345 a 10. 

R 2 



244 ARISTOTLE parti 

south : in the former position the comet does not attract the mois- 
ture to itself because the region is burnt up by the motion of the 
sun in it, and, when it is still further south, although there is plenty 
of moisture for the comet to attract, a question of angles (only a 
small part of the comet's circle being above the horizon) precludes 
the sight being reflected at the sun in this case, whether the sun be 
near its southern limit or at the summer solstice ; (3) of the comet's 
taking a tail when in a northerly position : the reason here being 
that a large portion of the comet's circle is above the horizon, and 
so the reflection of the sight is physically possible. Aristotle 
states objections, some of which apply to all, and others to some 
only, of the above views. Thus (1) the comet is not a planet, 
because all the planets are in the zodiac circle, while comets are 
often outside it ; (2) there have often been more than one comet at 
one time ; (3) if the tail is due to ' reflection ', and a comet has not 
a tail in all positions, it ought sometimes to appear without one ; 
but the five planets are all that we ever see, and they are often all 
of them visible above the horizon ; and, whether they are all visible, 
or some only are visible (the others being too near the sun), comets 
are often seen in addition. (4) It is not true that comets are only 
seen in the region towards the north and when the sun is near the 
summer solstice ; for the great comet which appeared at the time 
of the earthquake and tidal wave in Achaea [373/2 B.C.] appeared 
in the region where the sun sets at the equinox, and many comets 
have been seen in the south. Again (5) in the archonship of Eucles, 
the son of Molon, at Athens [427/6 B.C.], a comet appeared in the 
north in the month of Gamelion, when the sun was at the winter 
solstice, although, according to the theory, reflection of the sight 
would then be impossible. Aristotle proceeds : 

1 It is common ground with the thinkers just criticized and the 
supporters of the theory of coalescence that some of the fixed 
stars, too, take a tail ; on this we must accept the authority of the 
Egyptians (for they, too, assert it), and moreover we have ourselves 
seen it. For one of the stars in the haunch of the Dog got a tail, 
though only a faint one ; that is to say, when one looked intently 
at it, its light was faint, but when one glanced easily at it, it 
appeared brighter. 

' Moreover, all the comets seen in our time disappeared, without 
setting, in the expanse above the horizon, fading from sight by 



ch. χνπ ARISTOTLE 245 

slow degrees and in such a way that no astral substance, either 
one star or more, remained. For instance, the great comet before 
mentioned appeared in the winter of the archonship of Astaeus 
[37 '3/2 B -C-]> in clear and frosty weather, from the beginning of the 
evening ; the first day it was not seen because it had set before the 
sun, but on the following day it was visible, being the least distance 
behind the sun that allowed of its being seen at all, and setting 
directly ; the light of this comet stretched over a third part of the 
heaven with a great leap as it were {οίον άλμα), so that people 
called it a street. And it went back as far as the belt of Orion and 
there dispersed. 

1 Nevertheless Democritus for one stoutly defended his own theory, 
asserting that stars had actually been seen to remain on the disso- 
lution of comets. But in that case it should not have sometimes 
happened and sometimes failed to happen ; it should have hap- 
pened always. The Egyptians, too, say that conjunctions take 
place of planets with one another and of planets with the fixed stars ; 
we have, however, ourselves seen the star of Zeus twice meet one of 
the stars in the Twins and hide it, without any comet resulting.' 

Aristotle adds that this explanation of comets is untenable on 
general grounds, since, although stars may seem large or small, 
they appear to be indivisible in themselves. Now, if they were really 
indivisible, they would not produce anything bigger by coming in 
contact with one another ; therefore similarly, if they only seem 
indivisible, they cannot seem by meeting to produce anything bigger. 

Aristotle's own theory of comets explains them as due, much 
like meteors, to exhalations rising from below and catching fire 
when they meet that other hot and dry substance (also here called 
exhalation) which, being the first (i.e. outermost) portion of the 
sublunary sphere and in direct contact with the revolution of 
the upper (aethereal) part of the heavenly sphere, is carried round 
with that revolution and even takes with it part of the contiguous 
air. The necessary conditions for the formation of a comet, as 
distinct from a shooting star or meteor, are that the fiery principle 
which the motion of the upper heaven sets up in the exhalation 
must neither be so very strong as to produce swift and extensive 
combustion, nor yet so weak as to be speedily extinguished, but of 
moderate strength and moderate extent, and the exhalation itself 
must be ' well-tempered ' (ςνκρατος) ; according to the shape of the 
kindled exhalation it is a comet proper or the ' bearded ' variety 



246 ARISTOTLE part ι 

(ποογωι /ias). But two kinds of comets are distinguished. One is 
produced when the origin of the exhalation is in the sublunary- 
sphere ; this is the independent comet (καθ' iavrbv κομήτης). The 
other is produced when it is one of the stars, a planet or a fixed 
star, which causes the exhalation, in which case the star becomes a 
comet and is followed round in its course by the exhalation, just as 
haloes are seen to follow the sun and the moon. Comets are thus 
bodies of vapour in a state of slow combustion, moving either freely 
or in the wake of a star. Aristotle maintains that his view that 
comets are formed by fire produced from exhalations in the manner 
described is confirmed by the fact that in general they are a sign of 
winds and droughts. When they are dense and there are more 
of them, the years in which they appear are noticeably dry and 
windy ; when they are fewer and fainter, these characteristics are 
less pronounced, though there is generally some excess of wind 
either in respect of duration or of strength. He adds the following 
remarks on particular cases : 

' On the occasion when the (meteoric) stone fell from the air at 
Aegospotami, it was caught up by a wind and was hurled down in 
the course of a day; 1 and at that time too a comet appeared from 
the beginning of the evening. Again, at the time of the great comet 
[373/2 B.C., see pp. 244, 245 above] the winter was dry and arctic, 
and the tidal wave was caused by the clashing of contrary winds ; 
for in the bay the north wind prevailed, while outside it a strong 
south wind blew. Further, during the archonship of Nicomachus 
at Athens [341/0 B. c] a comet was seen for a few days in the 
neighbourhood of the equinoctial circle ; it was at the time of this 
comet, which did not rise with the beginning of the evening, that 
the great gale at Corinth occurred.' 

1 This appears to be the earliest mention of the meteoric stone of Aegospotami 
by any writer whose works have survived. The date of the occurrence was 
apparently in the archonship of Theagenides [468/7 B.C.]. The story that 
Anaxagoras prophesied that this stone would fall from the sun (Diog. L. ii. 10) 
was probably invented by way of a picturesque inference from his well-known 
theory that the fiery aether whirling round the earth snatched stones from the 
earth and, carrying them round with it, kindled them into stars (Aet. ii. 13. 3 ; 
D.G. p. 341 ; Vorsokratiker, i 2 , p. 307. 16), and that one of the bodies fixed in the 
heaven might break away and fall (Diog. L. ii. 12; Plutarch, Lysander 12; 
Vorsokratiker, i 2 , pp. 294. 29, 296. 34). Diogenes of Apollonia, too, a contempo- 
rary of Anaxagoras, said that along with the visible stars there are also stones 
carried round, which are invisible, and are accordingly unnamed ; ' and these 
often fall upon the earth and are extinguished like the stone star which made 
a fiery fall at Aegospotami' (Aet. ii. 13. 9; D. G. p. 342; Vorsokratiker, 
i 2 > P- 330. 5-8)• 



CH. χνπ ARISTOTLE 247 

It has been pointed out that Aristotle's account of comets held 
its ground among the most distinguished astronomers till the time 
of Newton. 1 

Passing to the subject of the Milky Way, 2 Aristotle again begins 
with criticisms of earlier views. The first opinion mentioned is 
that of the Pythagoreans, some of whom said that it was the path 
of one of the stars which were cast out of their places in the 
destruction said to have occurred in Phaethon's time ; while others 
said that it was the path formerly described by the sun, so that this 
region was, so to speak, set on fire by the sun's motion. But, 
Aristotle replies, if this were so, the zodiac circle should be burnt 
up too, nay more so, since it is the path not only of the sun but of 
the planets also. But we see the whole of the zodiac circle at one 
time or another, half of it being seen in a night ; and there is no 
sign of such a condition except at points where it touches the 
Milky Way. The remarkable hypothesis of Anaxagoras and 
Democritus is next controverted ; we have already (pp. 83-5) quoted 
Aristotle's criticisms. Next, a third view is mentioned according to 
which the Milky Way is ' a reflection of our sight at the sun ', just 
as comets had been declared to be. Aristotle refutes this rather 
elaborately. (1) If, he says, the eye, the mirror (the sun) and the 
thing seen (the Milky Way) were all at rest, one and the same part 

1 Ideler, Aristotelis Meteorologica, vol. i, p. 396. Yet Seneca (Nat. Quaes t. 
vii) had much sounder views on comets. He would not admit that they could 
be due to such fleeting causes as exhalations and rapid motions, as of whirlwinds, 
igniting them ; if this were their cause, how could they be visible for six months 
at a time (vii. 10. 1) ? They are not the effects of sudden combustion at all, but 
eternal products of nature (22. 1). Nor are they confined to the sublunary 
sphere, for we see them in the upper heaven among the stars (8. 4). If it 
is said that they cannot be ' wandering stars ' because they do not move in 
the zodiac circle, the answer is that there is no reason why, in a universe 
so marvellously constructed, there should not be orbits in other regions than 
the zodiac which stars or comets may follow (24. 2-3). It is true, he says, that, 
owing to the infrequency of the appearances of comets, their orbits have not 
as yet been determined, nay, it has not been possible even to decide whether 
they keep up a definite succession and duly appear on appointed days. In 
order to settle these questions, we require a continuous record of the appearances ' 
of comets from ancient times onwards (3. 1). When generation after generation 
of observers have accumulated such records, there will come a time when the 
mystery will be cleared up ; men will some day be found to show ' in what 
regions comets run their courses, why each of them roams so far away from the 
others, how large they are and what their nature ; let us, for our part, be content 
with what we have already discovered, and let our posterity in their turn contribute 
to the sum of truth (25. 7).' 2 Meteor, i. 8, 345 a 11 - 346 b io. 



248 ARISTOTLE 

of the reflection would belong to one and the same point of the 
mirror ; but if the mirror and the thing seen move at invariable 
distances from our eye (which is at rest), but at different speeds and 
distances relatively to one another, it is impossible that the same 
part of the reflection should always be at the same point of the 
mirror. Now the latter of the two hypotheses is that which corre- 
sponds to the facts, because the stars in the Milky Way and the sun 
respectively move at invariable distances from us, but at different 
distances and speeds in relation to one another ; for the Dolphin 
rises sometimes at midnight, and sometimes at sunrise, but the 
parts of the Milky Way remain the same in either case ; this could 
not be so if the Milky Way were a reflection instead of a condition 
'of the actual localities over which it extends. (2) Besides, how can 
the visual rays be reflected at the sun during the night? Aristotle's 
own explanation puts the Milky Way on the same footing as the 
second kind of comets, those in which the separation of the vapour 
which takes fire on coming into contact with the outer revolution is 
caused by one of the stars ; the difference is that what in the case 
of the comet happens with one star takes place in the case of the 
Milky Way throughout a whole circle of the heaven and the outer 
revolution. The zodiac circle, owing to the motion in it of the sun 
and planets, prevents the formation of the exhalations in that neigh- 
bourhood ; hence most comets are seen outside the tropic circles. 
The sun and moon do not become comets because they separate 
out the exhalation too quickly to allow it to accumulate to the 
necessary extent. The Milky Way, on the other hand, represents 
the greatest extent of the operation of the process of exhalation ; 
it forms a great circle and is so placed as to extend far beyond the 
tropic circles. The space which it occupies is filled with very great 
and very bright stars, as well as with those which are called l scat- 
tered ' (σποράδων) ; this is the reason why the collected exhalations 
here form a concretion so continuous and so permanent. The 
cause is indeed indicated by the fact that the brightness is greater 
in that half of the circle where it is double, for it is there that the 
stars are more numerous and closer together than elsewhere. 



XVIII 
HERACLIDES OF PONTUS 

The Pythagorean hypothesis of the revolution of the earth with 
the counter-earth, and of the sun, moon, and planets, about the 
central fire disappeared with the last representatives of the Pytha- 
gorean school soon after the time of Plato. The counter-earth was 
the first part of the system to be abandoned ; and it is suggested 
that this abandonment was due to the extension of the geographical 
horizon. Discoveries were made both to the east and to the west. 
Hanno, the Carthaginian, had made his great voyage of discovery 
beyond the Pillars of Hercules, and on the other (the eastern) side 
India became part of the known world. It would naturally be 
expected that, if journeys were made far enough to the east and 
west, points would be reached from which it should be possible 
to see the counter-earth, but, as neither the counter-earth nor the 
central fire proved in fact to be visible, this portion of the Pytha- 
gorean system had to be sacrificed. 1 

We hear of a Pythagorean system in which the central fire was 
not outside the earth but in the centre of the earth itself. Simplicius, 2 
in a note upon the passage of Aristotle describing the system of 'the 
Italian philosophers called Pythagoreans ' in which the earth revolves 
about the central fire and so ' makes day and night ', while it has 
the counter-earth opposite to it, adds that this is the theory of the 
Pythagoreans as Aristotle understood it, but that those who repre- 
sented the more genuine Pythagorean doctrine ' describe as fire at 
the centre the creative force which from the centre gives life to all 
the earth and warms afresh that part of it which has cooled down. 
. . . They called the earth a star, as being itself too an instrument of 
time. For the earth is the cause of days and of nights, since it makes 
day when it is lit up in that part which faces the sun, and it makes 

1 Gomperz, Griechische Denker, i 3 , pp. 97, 98 ; Schiaparelli, I firecursori di 
Copernico nelV antichita, pp. 22, 25. 

2 Simplicius on De caelo, p. 512. 9-20, Heib. 



250 HERACLIDES OF PONTUS parti 

night throughout the cone formed by its shadow. And the name 
of counter-earth was given by the Pythagoreans to the moon, just 
as they also called it " earth in the aether" (aiBepiav yr\v), both 
because it intercepts the sun's light, which is characteristic of the 
earth, and because it marks a delimitation of the heavenly regions, 
as the earth limits the portion below the moon.' 

It is no doubt attractive to suppose, as Boeckh τ does, that we 
have here a later modification of the system of Philolaus. But, 
as Martin 2 points out and Boeckh 3 admits, the earth in the system 
described by Simplicius is not in motion but at rest. For Simplicius, 
so far from implying that the earth rotates, thinks it necessary to 
explain how the Pythagoreans to whom he refers could, notwith- 
standing the earth's immobility, call it a ' star ' and count it, exactly 
as Plato does, among the ' instruments of time \ The fact is that 
the system, except for the detail of the term ' counter-earth ' being 
applied to the moon, agrees with the Platonic system as described 
in the Timaeus, and, as we have seen, there is nothing to suggest 
that Plato was acquainted with the Philolaic system at all ; he was 
rather basing himself upon the views of Pythagoras and the first 
Pythagoreans. 

A scholiast, writing on the same passage of Aristotle and 
describing the views of the Pythagoreans in almost the same 
words as those used by Simplicius, does, however, attribute motion 
to the earth. They put, he says, the fire at the centre of the 
earth. ' They said that the earth was a star as being itself too an 
" instrument ". The counter-earth for them meant the moon. . . . 
And this star [i. e. evidently the earth] by its motion (φζρομζνον) 
makes night and the day, because the cone formed by its shadow 
is night, while day is the illuminated part of it which is in the 
sun.' 4 The attribution of motion to the earth may be due to 
a misapprehension by the scholiast, just as Boeckh himself had 
at first assumed the earth's rotation to be indicated in the passage 
of Simplicius. 

However this may be, if the system of Philolaus be taken, and 
the central fire be transferred to the centre of the earth (the 

1 Boeckh, Das kosmische System des Platon, p. 96. 

2 Martin, Etudes sur le Time'e, ii, p. 104. 3 Boeckh, loc. cit. 
* Scholia in Aristotelem (Brandis), pp. 504 b 42 - 505 a 5. 



ch. χνπι HERACLIDES OF PONTUS 251 

counter-earth being also eliminated), and if the movements of the 
earth, sun, moon, and planets round the centre be retained without 
any modification save that which is mathematically involved by the 
transfer of the central fire to the centre of the earth, the daily revo- 
lution of the earth about the central fire is necessarily transformed 
into a rotation of the earth about its own axis in about 24 hours. 

All authorities agree that Heraclides of Pontus affirmed the daily 
rotation of the earth about its own axis ; but the Doxographi 
associate with this discovery another name, that of * Ecphantus 
the Pythagorean '. Thus we are told of Ecphantus that he asserted 
1 that the earth, being in the centre of the universe, moves about 
its own centre in an eastward direction '. l Again, * Heraclides of 
Pontus and Ecphantus the Pythagorean make the earth move, not 
in the sense of translation, but by way of turning as on an axle, 
like a wheel, from west to east, about its own centre.' 2 Who then 
is this Ecphantus, described in another place in Aetius as Ecphantus 
the Syracusan, one of the Pythagoreans ? His personality is even 
more of a mystery than that of Hicetas. The Doxographi, however, 
tell us of other doctrines of his; Hippolytus 3 devotes a short 
paragraph to him, between paragraphs about Xenophanes and 
Hippon, which shows that Theophrastus must have spoken of him 
at length. Some of his views were quite original, particularly on 
the subject of atoms. Holding that the universe was made up of 
indivisible bodies separated by void, he was the first to declare 
that the monads of Pythagoras were corporeal; he attributed to 
the atoms, besides size and shape, a motive force (δνι/αμι,ς) ; the 
atoms were moved, not by their weight or by percussion, but by 
a divine force which he called mind and soul. The universe was 
a type of this, and accordingly the divine motive force created it 
spherical. Now it is remarkable that Ecphantus's views all agree 
with Heraclides' so far as we know them ; Heraclides has the same 
divine force moving the universe, which he also calls mind and soul ; 
he has the same theory of atoms, which he calls masses 4 (ογκοή. 
And the two hold the same view about the rotation of the earth. 

1 Hippolytus, Refut. i. 15 {D. G. p. 566 ; Vors. i 2 , p. 265. 35). 

2 Aet. iii. 13. 3 (D. G. p. 378 ; Vors. i 2 , p, 266. 5). 3 Hippolytus, loc. cit. 
4 Galen, Histor. phil. 18 (D. G. p. 610. 22); Dionysius episcop. ap. Euseb., 

P.E. xiv. 23^ See Otto Voss, De Heraclidis vita et scriptis, p. 64; Tannery, 
Revue des Etudes grecques, x, 1897, pp. 134-6. 



252 HERACLIDES OF PONTUS parti 

Zeller observes, in addition, that the remark about the universe 
being made spherical reminds us of Plato. 1 Just as in the case of 
Hicetas, the natural conclusion is that the views attributed by the 
Doxographi to Ecphantus were expressed in a dialogue of Heraclides 
and put into the mouth of Ecphantus represented as a Pythagorean. 
Theophrastus may then have said something of this sort : ' Hera- 
clides of Pontus has developed the following theories, attributing 
them to a certain Ecphantus ' ; and this would account for the 
Doxographi citing the doctrines sometimes by the name of Heraclides, 
sometimes by the name of Ecphantus. 2 

Heraclides, son of Euthyphron, was born at Heraclea in Pontus, 
probably not many years later than 388 B.C. He is said to have 
been wealthy and of ancient family. He went to Athens not later 
than 364, and there met Speusippus, who introduced him into the 
school of Plato. Proclus, it is true, denied that he was a pupil 
of Plato, 3 but this was because Proclus resented his contradiction 
of the Platonic view of the absolute immobility of the earth in the 
centre of the universe. Diogenes Laertius, 4 Simplicius, 5 Strabo, 6 
and Cicero 7 leave us in no doubt on the subject; and we may 
also infer his relation to Plato from words of his own quoted 
elsewhere by Proclus, 8 according to which he was sent by Plato 
on an expedition to Colophon to collect the poems of Antimachus. 
Suidas 9 says that, during a journey of Plato to Sicily, Heraclides 
was left in charge of the school. After the death of Plato in 347, 
Speusippus was at the head of the school for eight years, and on 
his death in 338 B.C. Xenocrates was elected his successor, 
Heraclides and Menedemus, who were also candidates, being beaten 
by a few votes. 10 Heraclides then returned to his native town, 
where he seems to have lived till 315 or 310 B.C. While at 
Athens he is said to have attended the lectures of Aristotle also ; u 
but Diogenes' statement that he also 'heard the Pythagoreans' 

1 Zeller, i 5 , pp. 494, 495. 2 Tannery, loc. cit., p. 136. 

3 Proclus, in Tim. 281 Ε. 4 Diog. L. iii. 46, v. 86. 

5 Simpl. in Ar. Phys. iii. 4 (p. 202 b 36), p. 453. 29, Diels. 

6 Strabo, xii. 3. 1, p. 541. 

7 Cic. De nat. deor. i. 13. 34; De legg. iii. 6. 14; Tusc. Disp. v. 3. 8 ; De 
Divin.i. 23. 46. 8 Proclus, in Tim. 28 C 

9 Suidas, s. v. Ήρακλύδης. Zeller and Wilamowitz adduce confirmatory 
evidence. Voss alone disputes the statements ; for references see Voss, pp. 1 1-12. 
i0 Ind. Acad. Hercul. vi (Voss, p. 7). n Sotion in Diog. L. v. 86. 



ch. χνπι HERACLIDES OF PONTUS 253 

is no doubt incorrect ; for by that time the Pythagoreans had left 
Greece altogether. The words were probably interpolated in the 
passage of Diogenes by some one who inferred first-hand ac- 
quaintance with Pythagorean doctrines on the part of Heraclides 
from the fact, among others, that he wrote a book * concerning 
the Pythagoreans '} 

Diogenes Laertius tells us that Heraclides wrote works of the 
highest class both in matter and style. The remark is followed 
by a catalogue covering a very wide range of subjects, ethical, 
grammatical, musical and poetical, rhetorical, historical, with a note 
that there were geometrical and dialectical treatises as well. His 
dialogues are classified as (1) those which were by way of comedy, 
e.g. those on Pleasure and on Prudence, (2) those which were 
tragic, such as those on Things in Hades and on Piety, and (3) in- 
termediate in character, familiar dialogues between philosophers, 
soldiers, and statesmen. They were very varied and very persuasive 
in style, adorned with myth and full of imagination, so original as to 
make Timaeus describe their author as παραδοξολόγο? throughout, 
while Epicurus and the Epicureans, who attacked his physical 
theories, spoke of him as ' cramming his books with puerile stories \ 
There seems to have been more action in his dialogues than in 
Plato's ; 2 his prologues generally had nothing to do with what 
followed ; 3 there were usually a number of characters, and he 
was fond of introducing as interlocutors personages of ancient 
times. 4 He was much read and imitated at Rome, e.g. by Varro 
and Cicero; Cicero, for example, modelled upon Heraclides his 
dialogue De republica. Two of his dialogues at least, those ' On 
Nature ' and ' On the Heavens ', may have dealt with astronomy. 

He naturally had enemies, who not only impugned his doctrines 
but took objection to his personality. We are told that he was 
effeminate in dress and over-corpulent, so that he was called, 
not Ponticus, but Pompicus (Πομπίκός) ; his gait was slow and 
stately. 5 _ 

Several of the fragments of Heraclides recall passages in Plato. 
Thus Heraclides represents souls as coming down, for incarnation, 

1 Voss, pp. 12-13. 2 Ibid., pp. 26, 27. 

3 Proclus, in Plat. Parmenidem^ Book i, ad fin. 

4 Voss, p. 22. 5 Diog. L. v. 86. 



254 HERACLIDES OF PONTUS parti 

from regions in the heaven, which he places in or about the Milky- 
Way x (cf. the Phaedrus myth). The universe is a god ; so are the 
planets, the earth, and the heaven. 2 Other views of his about the 
universe and what it contains may also be referred to before we 
pass to the great discoveries in astronomy on which his fame rests. 
The universe is infinite ; 3 each star is also a universe or world, sus- 
pended in the infinite aether and comprising an earth, an atmosphere 
and an aether. 4 The moon is earth surrounded with mist. 5 Comets 
are clouds high in air lit up by the fire on high ; he accounts 
similarly for meteors and the like ; their different forms follow that 
of the cloud. 6 

We now pass to the first of Heraclides' great discoveries, that 
of the daily rotation of the earth about its axis. Besides the 
passages above quoted, in which ' Ecphantus ' is also credited with 
the discovery, we have the following clear evidence on the 
subject : 

' He (Aristotle) thought it right to take account of the hypothesis 
that both (i.e. the stars and the heaven as a whole) are at rest — 
although it would appear impossible to account for their apparent 
change of position on the assumption that both are at rest — because 
there have been some, like Heraclides of Pontus and Aristarchus, 
who supposed that the phenomena can be saved if the heaven 
and the stars are at rest while the earth moves about the poles of 
the equinoctial circle from the west (to the east), completing one 
revolution each day, approximately ; the ' approximately ' is added 
because of the daily motion of the sun to the extent of one degree. 
For of course, if the earth did not move at all, as he will later 
show to be the case, although he here assumes that it does for the 
sake of argument, it would be impossible for the phenomena to be 
saved on the supposition that the heaven and the stars are at 
rest.' 7 

* But Heraclides of Pontus supposed that the earth is in the 
centre and rotates (lit. ' moves in a circle ') while the heaven is 
at rest, and thought by this supposition to save the phenomena.' 8 

1 Heraclides of Pontus supposed that the earth moves about the 

1 Iamblichus in Stobaeus, Flor., p. 378, ed. Wachsmuth. 

2 Cicero, De nat. deor. i. 13. 34 (D. G. p. 541. 3-13). 
Aet. ii. 1. 5 (D. G. p. 328 b 4). 4 Aet. ii. 13. 15 (D. G. p. 343). 

Aet. ii. 25. 13 (D. G. p. 356). 6 Aet. iii. 2. 5 (D. G. pp. 366, 367). 

7 Simplicius on De caelo ii. 7 (289 b 1), pp. 444. 31 - 445. 5, Heib. 

* Ibid, (on c. 13, 293 b 30), p. 519. 9-1 1, Heib. 



ch. χνπι HERACLIDES OF PONTUS 255 

centre, while the heaven is at rest, and thought in this way to 
save the phenomena.' * 

* This would equally have happened [i. e. the stars would have 
seemed to be at different distances at different times instead of, 
as now, appearing to be always at the same distance, whether at 
rising or at setting or between these times, and the moon would 
not, when eclipsed, always have been diametrically opposite the sun, 
but would sometimes have been separated from it by an arc less 
than a semicircle] if the earth had a motion of translation ; but 
if the earth rotated about its centre while the heavenly bodies were 
at rest, as Heraclides of Pontus supposed, then (1), on the hypo- 
thesis of rotation towards the west, the stars would have been seen 
to rise from that side, while (2) on the hypothesis of rotation towards 
the east, (a) if it so rotated about the poles of the equinoctial circle 
(the equator), the sun and the other planets would not have risen 
at different points of the horizon [!], and, (b) if it so rotated about 
the poles of the zodiac circle, the fixed stars would not always have 
risen at the same points, as in fact they do ; so that, whether 
it rotated about the poles of the equinoctial circle or about the 
poles of the zodiac, how could the translation of the planets in 
the direct order of the signs have been saved on the assumption of 
the immobility of the heavens ? ' 2 

1 How can we, when we are told that the earth is wound round, 
reasonably make it turn round as well and give this as Plato's 
view ? Let Heraclides of Pontus, who was not a disciple of Plato, 
hold this opinion and move the earth round and round (κύκλω) ; 
but Plato made it unmoved/ 3 

The second great advance towards the Copernican system made 

by Heraclides was his discovery of the fact that Venus and Mercury 

revolve round the sun as centre. Some of the passages alluding to 

Heraclides' recognition of this fact import the later doctrine of 

epicycles ; but it is not difficult to eliminate this anachronism and 

to arrive at Heraclides' true theory. In some of the references 

the name of Heraclides is not mentioned. Vitruvius 4 describes the 

hypothesis thus : 

'The stars of Mercury and Venus make their retrograde motions 
and retardations about the rays of the sun, forming by their courses 
a wreath or crown about the sun itself as centre. It is also owing 
to this circling that they linger at their stationary points in the 
spaces occupied by the signs.' 

1 Schol. in Arist. (Brandis), p. 505 b 46-7. 

2 Simpl. on De caelo ii. 14 (297 a 2), pp. 541. 27 - 542. 2, Heib. 

3 Proclus, in Tim. 281 Ε. 4 Vitruvius, De architectura ix. 1 (4). 6. 



256 HERACLIDES OF PONTUS parti 

Next Martianus Capella 1 , who drew from Varro's work on astro- 
nomy, mentions the same hypothesis, but again without the name 
of its discoverer. 

' For, although Venus and Mercury are seen to rise and set daily, 
their orbits do not encircle the earth at all, but circle round the sun 
in a freer motion. In fact, they make the sun the centre of their 
circles, so that they are sometimes carried above it, at other times 
below it and nearer to the earth, and Venus diverges from the sun 
by the breadth of one sign and a half [45°]. But, when they are 
above the sun, Mercury is the nearer to the earth, and when they 
are below the sun, Venus is the nearer, as it circles in a greater 
and wider-spread orbit 

1 The circles of Mercury and Venus I have above described as 
epicycles. That is, they do not include the round earth within 
their own orbit, but revolve laterally to it in a certain way.' 

Cicero says that the courses of Venus and Mercury ' follow the 
sun as companions ', 2 but has nothing about their revolving round 
the sun. 

It is in Chalcidius 3 that we find the name of Heraclides con- 
nected with the revolution of the planets Mercury and Venus round 
the sun as centre ; but, like Adrastus in Theon of Smyrna, he 
erroneously imputes to Heraclides, as to Plato in the Timaeus, the 
machinery of epicycles. His words are : 

1 Lastly Heraclides Ponticus, when describing the circle of Lucifer 
as well as that of the sun, and giving the two circles one centre and 
one middle, showed how Lucifer is sometimes above, sometimes 
below the sun. For he says that the position of the sun, the moon, 
Lucifer, and all the planets, wherever they are, is defined by one 
line passing from the centre of the earth to that of the particular 
heavenly body. There will then be one straight line drawn from 
the centre of the earth showing the position of the sun, and there 
will equally be two other straight lines to the right and left of it 
respectively, and distant 50° from it, and ioo° degrees from each 
other, the line nearest to the east showing the position of Lucifer 
or the Morning Star when it is furthest from the sun and near the 
eastern regions, a position in virtue of which it then receives the 
name of the Evening Star, because it appears in the east at evening 
after the setting of the sun.' .... (And so on.) 

1 Martianus Capella, De nufitiis Philologiae et Mercurii, viii. 880, 882. 

2 Cicero, Somn. Scip. c. 4. 2. 

s Chalcidius, Timaeus, c. no, pp. 176-7, Wrobel. 



CH. χνπι HERACLIDES OF PONTUS 257 

Chalcidius only mentions Venus in this passage, but he has just 
previously indicated a similar relation between Mercury and the 
sun. Reading this passage and the explanation, illustrated by 
a figure, which follows, together with supplementary particulars 
given in a passage of Macrobius presently to be mentioned, we can 
easily realize Chalcidius's conception. According to this we are 
to suppose a point which revolves uniformly about the earth from 
west to east in a year. • This point is the centre of three concentric 
circles (epicycles) on which move respectively the sun (on the 
innermost), Mercury (on the middle circle), and Venus (on the 
outermost) ; the sun takes, of course, a year to describe its epi- 
cycle. 1 That the epicycle for the sun is wrongly imported into 
Heraclides' true system is confirmed by the next chapter of Chal- 
cidius, with its illustrative figure, where he imports epicycles into 
Plato's system also. According to him, Plato used, not one principal 
circle with three epicycles having as their common centre a point 
describing that principal circle, but three principal circles, each with 
one epicycle ; two circles, namely a principal circle and an epicycle, 
being used for each of the three bodies, the sun, Mercury, and Venus. 
But we know that in Plato's system the sun, Mercury, and Venus 
described three simple circles of which the earth is the centre. 
Hence the epicycles must be rejected altogether so far as Plato's 
system is concerned. Similarly, we must eliminate the sun's epi- 
cycle from the account of Heraclides' system, and we must suppose 
that he regarded Mercury and Venus as simply revolving in con- 
centric circles about the sun. 

The same contrast as is drawn by Chalcidius between Heraclides' 
system and Plato's system, as he represents them respectively, is 
drawn by Adrastus 2 between two possible theories, the authors of 

1 Chalcidius indicates (cc. 81, 109, and 112) that the sun's motion on its epi- 
cycle (which is from east to west) is in the contrary sense to the motion (from 
west to east) of Mercury and Venus on their epicycles respectively (cf. Adrastus 
in Theon of Smyrna, p. 175, 13-15, who says that the motion of the sun and moon 
on their epicycles is in the sense of the daily rotation from east to west, while the 
motion of the five planets on their epicycles is in the opposite sense). The 
commentators did not fail to see in this fact a possible explanation of Plato's 
remark that Mercury and Venus have 'the contrary tendency to the sun' (Chal- 
cidius, c. 109, p. 176) ; and the explanation would be quite satisfactory z/Plato 
could be supposed to have been acquainted with the theory of epicycles (cf. 
pp. 165-9 above). 

2 Adrastus in Theon of Smyrna, pp. 186. 17- 187. 13. 
1410 s 



258 HERACLIDES OF PONTUS parti 

which he does not specify. The first possibility corresponds to 
Chalcidius's version of Plato's system ; only Hipparchus's epicycles 
are, in agreement with Eudoxus's theory of spheres, represented 
by ' solid ' spheres as distinct from ' hollow '. We are to conceive, 
in the plane of the ecliptic, three concentric circles with the earth as 
common centre ; on each circle there moves, in one and the same 
direction, the centre of an immaterial sphere at such speed that the 
centre of the earth and these three centres are always in a straight 
line. As the plane of the ecliptic cuts the three immaterial spheres, 
this determines three circles which, with Hipparchus, we distin- 
guish from the principal circles as epicycles. The sun moves on 
the epicycle of the circle nearest the earth, Mercury on that of the 
next, Venus on that of the outer circle. This is, therefore, precisely 
the Platonic system as conceived by Chalcidius. The second possi- 
bility, says Adrastus, is that the three principal circles may coalesce 
into one. Thus the three epicycles are reduced to sections of three 
concentric spheres, and the whole system of these spheres revolves 
about the earth, their common centre describing a circle about the 
earth. Here we have Heraclides' system as described by Chalcidius ; 
but Adrastus's version is better, in that, evidently relying on an older 
source, he hints that what moves on the main circle is not an 
immaterial point but the 'true solid sphere of the sun'; that is to 
say, it is only Mercury and Venus which move on epicycles, i.e. in 
circles about the sun as centre. 1 

Martin 2 exposed the error of those who inferred from the passage 
of Macrobius already alluded to that the Egyptians were acquainted 
with the fact thus stated by Heraclides. Macrobius observes that 
Cicero, in placing the sun fourth in the order of the planets reckon- 
ing from the earth, i.e. after the moon, Venus, and Mercury, followed 
the order adopted by the Chaldaeans and Archimedes, while 

1 Plato followed the Egyptians, the parents of all branches of 
philosophy, who, while placing the sun between the moon and 
Mercury, yet have detected and enunciated the reason why the sun 
is believed by some to be above Mercury and above Venus ; for 
neither are those who hold this view far from the apparent 
truth.' 3 . . . 

1 Hultsch, ' Das astronomische System des Herakleides von Pontos' vajahrb. 
fur class. Philologie, 1896, pp. 305-16. 

8 Martin, Etudes sur le Timee, ii, pp. 130-3. Cf. Boeckh, Das kosmische 
System des Platen^ pp. 142, 143. 8 Macrobius, In somn. Scip. i. 19. 2. 



ch. χνιπ HERACLIDES OF PONTUS 259 

Then, after explaining that Saturn is as far from Jupiter as is 
indicated by the difference between their periods, 30 years and 
12 years respectively, and again, that Jupiter's distance from Mars 
corresponds to the difference between their periods of 12 and 2 years 
respectively, he observes that Venus is so much below Mars as 
corresponds to the shorter period of Venus, one year, while Mercury 
is so near to Venus, and the sun to Mercury, that they all describe 
their orbits in one year, more or less, so that, as Cicero says, Venus 
and Mercury are companions of the sun. There was, therefore, no 
dispute about the order of the superior planets, Saturn, Jupiter, and 
Mars, nor about the relative position of the moon as the lowest 
of all ; 

1 But the proximity of the three others which are the nearest to 
one another, namely Venus, Mercury, and the sun, has caused 
uncertainty as regards their order, though only in the minds of 
others, not of the Egyptians ; for the true relation did not escape the 
penetration of the Egyptians, and it is as follows. The circle on 
which the sun moves [' circulus, per quern sol discurrit' = the sun's 
epicycle] is lower than, and encircled by, the circle of Mercury ; 
above the circle of Mercury, and including it, is the circle of Venus ; 
hence it is that, when the two planets are describing the upper 
portions of their circles, they are regarded as placed above the sun, 
but when they are traversing the lower portions of their circles, the 
sun is considered to be superior to them.' 1 . . . 

Macrobius's main object may have been to put the Egyptians on 
a level with the Chaldaeans, the oldest cultured Asiatics. 2 But, 
though the Chaldaeans arranged the planets in an order different 
from that adopted by Plato, the idea of Mercury and Venus revolv- 
ing round the sun was certainly not Chaldaean but Greek, and 
originated with Heraclides. If Macrobius really intended to attri- 
bute Heraclides' discovery to the Egyptians, it must be because 
the theory had perpetuated itself as a tradition of the Alexandrine 
astronomers anterior to our era. 3 And if the Egyptians had 
really regarded Mercury and Venus as being in the relation of 
satellites to the sun, it is not easy to understand why they placed 
Mercury and Venus above the sun, since they might equally well 
have placed them below it. 

Hultsch explains the evolution of the Heraclides-epicyclic system 

1 Macrobius, In somn. Scift. i. 19, 5-6. 2 Hultsch, loc. cit. 

3 Tannery, Recherches sur Fhistoire de Γ astronomie ancienne y pp. 260, 261. 

S % 



26o HERACLIDES OF PONTUS parti 

in the following way. The axial rotation of the earth was rejected 
by Hipparchus. Hence the occasion, for some one living after 
Hipparchus's time, of modifying Heraclides' system and grafting 
on to it the theory of epicycles. Or perhaps the post-Hipparchian 
inventor of the Heraclides-epicyclic blend wished to oppose to some 
enthusiastic champion of Hipparchus the authority of Heraclides, 
but could not get rid of epicycles. 

The next question which arises is this. Having made Mercury 
and Venus revolve round the sun as satellites, did Heraclides 
proceed to draw the same inference with regard to the other, the 
superior, planets ? When it was once laid down that all the five 
planets alike revolved round the sun, and this hypothesis was com- 
bined with that of the revolution of the sun round the earth as 
centre, the result was the system of Tycho Brahe, with the improve- 
ment, already made by Heraclides, of the substitution of the daily 
rotation of the earth for the daily revolution of the whole system 
round the earth supposed at rest. Schiaparelli, who added to his 
first tract, I precursori di Copernico nell ' antichita, a further ex- 
tremely elaborate study 1 dealing at length with the above question 
among others, came to the conclusion that it was probably Hera- 
clides himself who took the further step of regarding all the five 
planets alike as revolving round the sun, but that, if it was not 
Heraclides, it was at all events some contemporary of his who did 
so. This conclusion represents a certain change of view on the 
part of Schiaparelli after the date of / precursori, where he says, 
' it appears that Heraclides Ponticus, as the evidence cited indicates, 
limited to Venus and Mercury the revolution round the sun, and it 
seems that he retained the earth as the centre of the movements 
of the superior planets \ 2 Schiaparelli's later view is based upon 
presumption rather than upon direct evidence, which indeed does 
not exist. His argument is a tour de force, but, although opinions 
will differ, I for my part think that he trusts too much to the testi- 
mony of late writers as to the supposed very early discovery of the 
machinery of eccentrics and epicycles, and his case does not seem 
to me to be made out. 

1 Schiaparelli, Origine del sistema planetario eliocentrico presso i Greet, 1898 
(in Memorie del R. Istituto Lombardo di scienze e lettere, vol. xviii, pp. 61 sqq.j. 

2 Schiaparelli, / precursori, pp. 27, 28. 



CH. χνπι HERACLIDES OF PONTUS 261 

Schiaparelli's arguments are, however, well worthy of considera- 
tion, and I will represent them as completely and fairly as I can. 
Having hit upon the hypothesis of the revolution of Mercury and 
Venus round the sun, and not the earth, as centre, Heraclides had 
found a possible explanation of the varying degrees of brightness 
shown by the two inferior planets and of the narrow limits of their 
deviation from the sun ; he would also easily see that the hypo- 
thesis gave a solution of the difficulty of the stationary points and 
the retrogradations in the case of these planets. Eudoxus had 
tried to solve the latter difficulty by ingenious and elegant combina- 
tions of concentric spheres ; but he only succeeded with Jupiter and 
Saturn. Callippus went further on the same lines and succeeded 
to a certain extent with Mars ; probably, too, he came nearer to 
accounting for the movements of Mercury and Venus. The most 
formidable objection to the explanation of the planetary move- 
ments by means of concentric spheres was the fact that, on this 
hypothesis, the distance of each planet from the earth, and conse- 
quently its brightness, should be absolutely invariable, whereas 
mere ocular observation sufficed to prove that this is not so. This 
difficulty was, as we have seen, very early realized ; Polemarchus, 
a friend of Eudoxus himself, was aware of it, but tried to make out 
that the inequality of the distance was negligible and of no account 
in comparison with the advantage of having all the spheres about 
one and the same centre ; Aristotle, too, in his Physical Problems 
(now lost) discussed the same difficulty. 1 The first who tried to get 
over the difficulty was Autolycus of Pitane, the author of the tract 
On the moving sphere, but even he was not successful. 2 Now Hera- 
clides, departing altogether from the system of spheres, to which 
the Aristotelian school doggedly adhered, and adopting a system of 
circles more akin to Pythagorean ideas, had suggested a sufficient 
explanation with regard to Venus and Mercury ; and, as Mars was 
seen, equally with Venus, to vary in apparent size and brightness, 
it was natural for the same school of thought to try to find an 
explanation of the similar phenomena with regard to Mars on their 
lines as opposed to those which found favour with Aristotle. 

Now, with regard to Mars, it would be seen that the times of its 

1 Sosigenes in Simplicius on De caelo (293 a 4), p. 505. 21-7, Heib. 

2 Ibid., p. 504. 22-5, Heib. 



202 HERACLIDES OF PONTUS parti 

greatest brightness corresponded with the times when it was in 
opposition and not in conjunction ; that is to say, it is brightest 
when it occupies a position in the zodiac opposite to the sun ; it 
must therefore be nearest the earth at that time, and consequently 
the centre of its orbit cannot be the centre of the earth, but must 
be on the straight line joining the earth to the sun. The analogy 
of Venus and Mercury might then suggest that perhaps Mars, too, 
might revolve round the sun. I do not attach much importance in 
this connexion to a passage from Theon of Smyrna quoted by 
Schiaparelli. Theon, in the passage contrasting two hypotheses 
(the supposed Platonic and supposed Heraclidean) with regard to 
the movements of Venus and Mercury, adds : 

' And one might suspect that this [the Heraclidean view] repre- 
sents the truer view of their relative position and order, the effect 
of it being to make this region the abode of the animating principle 
in the universe, regarded as a living thing, the sun being as it were 
the heart of the All in virtue of its great heat and in consequence 
of its motion, its size, and its connexion with the bodies about it. 
For in animate beings the centre of the thing, that is, of the animal 
as animal, is different from the centre of it regarded as a magnitude ; 
thus with ourselves as men and living beings one centre is the 
region about the heart, the centre of the vital principle . . . the other 
is that of the body as a magnitude. . . . Similarly, if we may extend 
to the greatest, noblest and divine the analogy of the small, insig- 
nificant and mortal, the centre of the universe as a magnitude is 
the region about the earth which is cold and destitute of motion ; 
while in the universe as universe and living thing the region about the 
sun is the centre of its animating principle, the sun being as it were 
the heart of the All, which is also, as we are told, the starting-point 
whence the soul proceeds to permeate the whole body spread over 
it from the extremities inwards.' 1 

The argument of Theon seems rather to be offered as a plausible 
defence of the new theory of Venus and Mercury as satellites of 
the sun, after the event as it were, than as an a priori ground for 
putting forward that hypothesis or for extending it to Mars and 
the other superior planets. 

When the possibility of Mars revolving round the sun came to 

1 Theon of Smyrna, pp. 187. 13 - 188. 7. Cf. Plutarch, Defac. in orbe lunae, 
c I5> P• 928 B, c ; Macrobius, In somn. Scip. i. 20. 1-8. 



CH. XVIII 



HERACLIDES OF PONTUS 



263 



be considered, it would be at once obvious that the precise hypo- 
thesis adopted for Mercury and Venus would not apply, because 
the circles described by those planets about the sun are relatively 
small circles and are entirely on one side of the earth, whereas the 
circle described by Mars comprehends the earth which is inside it. 
The next possibility that would present itself would be that the 
planet might move uniformly round an eccentric circle of some 
kind, a circle passing round the earth but with some other point 
not the earth as centre. Suppose Ε is the earth, fixed at the centre 
of the universe, QR an eccentric circle with centre O. Draw the 
diameter QR through E, 0. Then Q represents the perigee of 




Fig. 12. 

a planet moving on the eccentric circle. In opposition, therefore, 
Mars will be at Q, and the sun will be opposite to it, i.e. at some 
point on ER. If now the oppositions always occurred in the same 
place in the zodiac, i.e. in the same direction EQ, this hypothesis 
would explain the differences of brightness. But the oppositions 
do not always take place in the same direction ; they may take 
place at any part of the zodiac. Consequently, the direction of 
opposition is not constant, as EQ, but the diameter RQ must 
move round the centre Ε in such a way that the perigeal point Q 
is always opposite to the sun. Therefore Q, the point of opposition, 
revolves round Ε in the space of a year along the ecliptic in the 
direct order of the signs. _ Hence 0> the centre of the eccentric, also 



264 HERACLIDES OF PONTUS parti 

revolves round Ε in a year in such a way that it is always in the 
direction of the sun. We suppose, therefore, that the whole 
eccentric circle moves bodily round Ε as centre, as if it were 
a material disc attached to Ε as a sort of hinge. If now we 
suppose Mars to move uniformly round the circumference of the 
eccentric in the inverse order of the signs, completing the circuit 
from perigee to perigee, or from apogee to apogee, in a time equal to 
the period of its synodic revolution, the opposition will occur at the 
right places and the brightness will then be greatest. Further 
(and this is the most important point) if the distance EO (the 
1 eccentricity ') is chosen in the proper ratio to the radius OR, the 
irregular movements of the planet, its stationary positions, and its 
retrogradations will be explained also (this would be clear to any 
one who was enough of a geometer, though the corresponding facts 
are easier to see when the hypothesis is that of epicycles). By 
means of observations it would be possible to deduce the ratio 
of the radius to the ' eccentricity ', but not their absolute magni- 
tudes. But the centre Ο is always in the direction of the sun ; 
it only remained to fix its distance (EO). The natural thing in 
the case of Mars would be to make the material sun the centre, just 
as had been done with the epicycles of Venus and Mercury. The 
use of ideal points as centres for epicycles and eccentrics was no 
doubt first thought of, at a later stage, by some of the great 
mathematicians such as Apollonius. 

The next link in Schiaparelli's chain of argument is the fact 
that the same movement as is represented by movable eccentrics 
of the sort just described can equally well be represented by means 
of epicycles, a fact which is proved by Theon of Smyrna and 
others. Let us then see how the motion of Mars, as above repre- 
sented by means of a movable eccentric, can be represented by 
means of an epicycle. Let Figure 13 (a) represent a movable 
eccentric, Ε being the earth, *S the centre of the eccentric which 
moves round the circle SS' in the direction shown by the arrow, 
in such a way that ES is always in the direction of the sun and 
moves in the direct order of the signs. Let CC be the eccentric 
with centre 5. Produce ES to meet the eccentric in C, which will 
then be the position of the apogee of the eccentric. Let the planet 
be then at the point D describing the circle CC in the inverse 



CH. XVIII 



HERACLIDES OF PONTUS 



26s 



order of the signs. The angle CSD or the arc CD reckoned from 
the apogee in the inverse order of the signs will be the argument 
of the anomaly 1 or shortly the anomaly ; the planet will be seen 
from the earth in the direction ED. 

Now [Fig. 13 (b)], on the hypothesis of the epicycle, let Θ be the 
centre of the earth. About Θ as centre describe the circle X5" equal 
to the eccentric circle of the other figure, and draw the radius Θ^ 
parallel (and equal) to SD in the other figure. Take X as the centre 
of the epicycle, and about it describe the circle Δ Δ' equal to the 
circle SS' in the other figure. If we produce Θ2 to Κ, Κ will 
be at the moment the apogee of the epicycle. Make the angle 




(A) (B) 

Fig. 13. 

ΚϋΔ equal to the anomaly (i. e. the angle CSD in the other figure) 
but reckoned in the opposite sense (i.e. in the direct order of the 
signs). Suppose then that the planet is at Δ and seen from the 
earth in the direction θ J. 

In the triangles ESD, Θ2Δ, DS is equal and parallel to 2Θ, 
and the angles DSE, Θ2Δ are equal ; therefore ES, ΔΊΖ are 
parallel. But ES, Δ2 are also equal ; therefore the two sides 
ES, SD are equal to the two sides ΔΣ, 2Θ respectively. And 
the included angles are equal ; therefore the triangles ESD y Δ%Θ 
are equal in all respects. And, since the two sides ES, Δ% are 
equal and parallel, and the sides SD, Θ2 are also equal and 
parallel, it follows that the third sides ED, ΘΔ will be equal and 
parallel, i.e. the planet will be seen in the same direction and at 
the same distance under either hypothesis. 



2,66 HERACLIDES OF PONTUS parti 

The conditions necessary in order that this may be true at any 
instant are two : (i) the radii SD, Θ Ζ of the eccentric and the 
deferent circle respectively must always remain parallel ; (2) the 
anomaly CSD in the eccentric must be equal to the anomaly ΚΖΔ 
in the epicycle, while the anomaly must in the first case be reckoned 
in the inverse order, and in the second case in the direct order of the 
signs. It is evident also that the proof still holds if, instead of 
making the radii of the two circles in each hypothesis equal, we 
suppose them proportional only and change the dimensions of 
either figure as we please. 

It is clear why the Greek mathematicians preferred the epicycle 
hypothesis to the eccentric. It was because the former was applic- 
able to all cases ; it served for the inferior as well as the superior 
planets, whereas the eccentric hypothesis, as then conceived, would 
not serve for the inferior planets ; moreover, the epicycle hypothesis 
enabled the phenomena of the stationary points and retrograda- 
tions to be seen almost by simple inspection, whereas on the 
eccentric hypothesis a certain amount of geometrical proof would 
be necessary to enable the effect in this respect to be understood. 
But it will be observed that in the above figures the motion of S 
round the circle S f S may be the motion of the material sun in its 
orbit but, when this is so, the point X which is the centre of the 
epicycle in the other case is not a material but an ideal point. 
Hence, before geometers had fully developed the theory of revo- 
lution about ideal points, the eccentric hypothesis was the only 
practicable way of representing the movements of the superior 
planets, Mars, Jupiter, and Saturn. 

Now we infer from a passage of Ptolemy * that, while Apollonius 
understood the theory of epicycles in all its generality, he only 
knew of the particular class of eccentrics in which the movable 
centre of the eccentric moves at an angular speed equal to that 
of the sun describing its orbit about the earth. The description 
by Apollonius of the two hypotheses is in these words : 

(1) The epicycle hypothesis: 'Here the epicycle's advance in 
longitude is in the direct order of the signs round the circle con- 
centric with the zodiac, while the star moves on the epicycle about 
its centre at a speed equal to that of the anomaly and in the direct 
1 Ptolemy, Syntaxis xii. 1 (vol. ii, pp. 450. 10-17, 451. 6-14, Heib.). 



CH. χνπι HERACLIDES OF PONTUS 267 

order of the signs in that part of the circumference of the epicycle 
which is furthest from the earth.' 

(2) The eccentric hypothesis : * This is only applicable to the three 
pla7iets which can be at any angular distance whatever from the sun y 
and here the centre of the eccentric circle moves about the centre 
of the zodiac in the direct order of the signs and at a speed eqtial 
to that of the sun, while the star moves on the eccentric about its 
centre in the inverse order of the signs and at a speed equal to 
that of the anomaly.' 

What makes Apollonius say that the eccentric hypothesis is not 
applicable to the inferior planets is the fact that, in order to make it 
apply to them, we should have to suppose the circle described by the 
centre of the eccentric to be greater than the eccentric circle itself. 

The object of the passage of Ptolemy is to explain the stationary 
points and retrogradations on either hypothesis, and he reproduces 
in his own form two propositions which, he says, had been proved 
* by other mathematicians as well as by Apollonius of Perga with 
reference to one of the anomalies, the anomaly in relation to the 
sun.' It is from the passage in question that it has commonly been 
inferred that Apollonius of Perga was the inventor of epicycles. 
I agree, however, with Schiaparelli that, if we read the passage 
carefully, we shall find that it does not imply this. It is at least 
as easy to infer from the language of Apollonius that, in the case of 
the epicycle-hypothesis at all events, he was only stating formally 
what was already familiar to those conversant with the subject. 

Now the eccentric hypothesis, which is, in the proposition with 
regard to it proved by Apollonius, limited to the particular case 
of the three superior planets, was evidently generalized at or 
before the time of Hipparchus. This is clear from passages of 
Ptolemy and Theon of Smyrna quoted by Schiaparelli. (1) Ptolemy 
says that Hipparchus was the first to point out that it is necessary 
to explain how there are two kinds of anomaly in the case of each 
of the planets, the solar (ή πάρα τον ήλων ανωμαλία) and the 
zodiacal, or how the retrogradations of each planet are unequal 
and of such and such lengths, whereas all other mathematicians 
had based their geometrical proofs on the assumption that the 
anomaly and the retrogradation were one and the same respec- 
tively. Hipparchus added that these phenomena were not accounted 



268 HERACLIDES OF PONTUS parti 

for either by eccentric circles, or by circles concentric with the 
zodiac carrying epicycles, or even by a combination of both 
hypotheses. 1 (%) ' Hipparchus says it is worthy of investigation 
by mathematicians why on two hypotheses so different from one 
another, that of eccentric circles and that of concentric circles with 
epicycles, the same results appear to follow.' 2 A further allusion 
to the same remark of Hipparchus shows that the identity of the 
results following from the two hypotheses was shown with regard 
to the sun, 3 which is the case for which Adrastus proved it. 4 
Again, Theon of Smyrna says that ' Hipparchus prefers the hypo- 
thesis of the epicycle which he claims as his own, asserting that 
it is more natural that all the heavenly bodies should be properly 
balanced, and connected together in the same way, about the 
centre of the universe ; and yet, because he was not sufficiently 
equipped with physical knowledge, even he did not know for 
certain which is the natural and therefore true movement of the 
planets and which the incidental and apparent ; but he, too, supposes 
that the epicycle of each planet moves on the encentric circle and 
the planet on the epicycle'. 5 (3) In a famous passage where 
Simplicius reproduces a quotation by Alexander from Geminus or 
Posidonius (if Geminus was actually copying Posidonius) we read, 
' Why do the sun, moon, and planets appear to move irregularly ? 
Because, whether we suppose that their circles are eccentric or that 
they move on epicycles, their apparent irregularity will be saved ; 
and it will be necessary to go further and consider in how many ways 
these same phenomena are capable of being explained, in order that 
our theory of the planets may agree with that explanation of the 
causes which proves admissible.' 6 

The theory of eccentrics had therefore been generalized by 
Hipparchus's time, but with Apollonius was still limited to the case 
of the three superior planets. This indicates clearly enough that 
it was invented for the specific purpose of explaining the movements 
of Mars, Jupiter, and Saturn about the sun, and for that purpose 
alone. Who then took this step in the formulation of a system 

1 Ptolemy, Syntaxis ix. 2 (vol. ii, pp. 210. 19 -211. 4, Heib.). 

2 Theon of Smyrna, p. 166. 6-10. 3 Ibid., p. 185. 13-19. 
4 Ibid., pp. 166. 14-172. 14. 6 Ibid., p. 188. 15-24. 
6 Simplicius in Phys., p. 292. 15-20, ed. Diels. 



CH. χνπι HERACLIDES OF PONTUS 269 

which is the same as that of Tycho Brahe ? Tannery ] thinks it was 
Apollonius, and in that case Apollonius, coming after Aristarchus 
of Samos, would be exactly the Tycho Brahe of the Copernicus of 
antiquity. 

Schiaparelli, however, as I have said above, will have it that 
it was not Apollonius, but Heraclides or some contemporary of his, 
who took the final step towards the Tychonic system. In order 
to prove this it is necessary to show that epicycles and movable 
eccentric circles were both in use by Heraclides , time, and Schia- 
parelli tries to establish this by quotations from Geminus, Proclus, 
Theon of Smyrna, Chalcidius, and Simplicius ; but it is here that 
he seems to me to fail. The passages cited are as follows. 

(1) Geminus : ' It is a fundamental assumption in all astronomy 
that the sun, the moon, and the five planets move in circular 
orbits at uniform speed in a sense contrary to that of the 
universe. For the Pythagoreans, who were the first to apply 
themselves to investigations of this kind, assumed the movements 
of the sun, the moon, and the five planets to be circular and 
uniform. They would not admit, with reference to things divine 
and eternal, any disorder such as would make them move at one 
time more swiftly, at one time more slowly, and at another time 
stand still, as the five planets do at their so-called stationary points. 
For such irregularity of motion would not even be expected of 
a decent and orderly man in his journeys. With men, of course, 
the necessities of life are often causes of slowness and swiftness ; but 
with the imperishable stars it is not possible to adduce any cause 
of swiftness or slowness. Accordingly, they proposed the problem, 
how the phenomena could be accounted for by means of circular 
and uniform movements.' 2 Geminus goes on, it is true, to explain 
why the sun, although moving at uniform speed, describes equal 
arcs in unequal times, and explains the fact by assuming the sun 
to move uniformly in an eccentric circle, i.e. a circle of which the 
earth is an internal point but not the centre. But there is nothing 
to suggest that this was the Pythagorean answer to the problem. 
Geminus says ' We shall give the explanation as regards the other 

1 Tannery, Recherches sur Vhistoire de V astrono7?iie ancienne, c. 14, pp. 245, 

253-9. 

2 Geminus, Isagoge, c. 1. 19-21, p. 10. 2-20, ed. Manitius. 



zjo HERACLIDES OF PONTUS parti 

stars in another place ; but we will show at once with regard to the 
sun how . . . .' 

{%) Theon of Smyrna says, quoting Adrastus : l * The apparent 
intricacy of the motion of the planets is due to the fact that they 
seem to us to be carried through the signs of the zodiac in circles 
of their own, being fixed in spheres of their own and moved along 
the circles, as Pythagoras was the first to observe, a certain intricate 
and irregular movement being thus incidentally grafted on to their 
simple and uniform motion, which remains the same.' 

(3) Chalcidius says: 2 'Yet all the planets seem to us to move 
unequally and some even to show disordered movements. What 
then shall we give as the explanation of this erroneous supposition ? 
That mentioned above, which was also known to Pythagoras, 
namely that, while they are fixed in their own spheres and so 
carried round, they appear, owing to our feebleness of vision, to 
describe the circle of the zodiac' 

Schiaparelli adds : ' We cannot attribute any historical value to 
this notice unless we admit that by " Pythagoras" are to be under- 
stood those same Pythagoreans of whom Geminus speaks. And 
it would follow that those Pythagoreans had explained the irregu- 
larity of the planetary movements by means of the combination 
of two circular movements, one with the earth as centre, the other 
having its centre outside the earth (eccentric or epicycle).' But 
there is nothing whatever in these passages to suggest eccentrics 
or epicycles. Theon follows up his remark by referring to the 
combination of movements as explained by Plato in the Titnaeus, 
i. e. the supposition that, while the sun, moon, and planets have 
an independent circular movement of their own in the zodiac about 
the earth as centre, they also share in the movement of the fixed 
stars (the daily rotation about the axis of the universe). The 
passage of Chalcidius seems to mean the same thing. Martin 
interprets the passages of Geminus and Chalcidius as saying that 
Pythagoras denied the irregularity of the movement of the stars 
called planets, considering it an optical illusion. 3 Zeller observes 
that the passage of Theon indicates that the early Pythagoreans 

1 Theon of Smyrna, p. 150. 12-18. 

2 Chalcidius, Timaeus, c. 77, 78, pp. 145, 146, ed. Wrobel. 

3 Martin, Etudes sur le Ti?nie, ii, p. 120. 



ch. χνιπ HERACLIDES OF PONTUS 271 

developed the doctrine of Anaximander into a theory of spheres 
carrying round stars which are made fast to them, and that this 
is confirmed by the occurrence of the same conception in Parmenides 
and Plato. Whether all the stars are carried by spheres of their own, 
i.e. hollow spheres, or only the fixed stars are carried by one sphere, 
while the planets, as with Plato, are fixed on hoop-like circles, is not 
clear. But Zeller rejects altogether the view that the Pythagoreans 
assumed eccentrics and epicycles as not only unsupported by trust- 
worthy evidence but as inconsistent with the whole development of 
the old astronomy. 1 

But we have not done with the evidence cited by Schiaparelli. 

(4) Proclus says 2 : 'The hypotheses of eccentrics and epicycles com- 
mended themselves also, so history tells us, to the famous Pythagoreans 
as being more simple than all others — for it is necessary in dealing 
with this question, and Pythagoras himself encouraged his disciples, 
to try to solve the problem by means of the fewest and most simple 
hypotheses possible.' This passage, as Schiaparelli says, attributes 
the first idea of movable eccentrics as well as of epicycles to the Pytha- 
goreans. But it has tobe considered along with a passage of Simplicius 
which Schiaparelli regards as the most important notice of all ; 

(5) Simplicius says, after speaking of the system of concentric 
spheres : ' Later astronomers then, rejecting the hypothesis of 
revolving spheres, mainly because they do not suffice to explain 
the variations of distance and the irregularity of the movements, 
dispensed with concentric spheres and assumed eccentrics and 
epicycles instead — if indeed the hypothesis of eccentric circles 
was not invented by the Pythagoreans, as some tell us, in- 
cluding Nicomachus and Iamblichus who followed him.' 3 This 
passage, it is true, may indicate that it was only eccentric circles, 
and not epicycles also, which the Pythagoreans discovered ; but 
Schiaparelli regards it as conclusive with reference to movable 
eccentrics. Unfortunately, he has not allowed for the fact that 
it was the habit of the neo- Pythagoreans to attribute, so far as 
possible, every discovery to the Pythagoreans, and even to Pytha- 
goras himself. The evidence of Nicomachus would therefore 

1 Zeller, i 5 , p. 415 n. 

2 Proclus, Hypotyfiosis astronomicarum positionum, c. 1, § 34, p. 18, ed. 
Manitius. 8 Simplicius on De caelo, p. 507. 9-14, Heib. 



%η% HERACLIDES OF PONTUS parti 

be worthless even if it could not easily be accounted for ; but, as 
Hultsch says, 1 the statement is easily explained as a reminiscence 
of the Pythagorean central fire, for of course in that system each 
planet moved in a circle about the central fire as centre and, as the 
earth also moved round the same central fire, the orbit of the planet 
would be eccentric relatively to the earth. The passage of Proclus 
may be based on the authority of Nicomachus ; or it may be a case 
of a wrong inference, thus : the Pythagoreans sought the simplest 
hypothesis because they held that that would be the best ; the 
simplest is that of eccentrics and epicycles ; therefore the Pytha- 
goreans would naturally think of that hypothesis. 

But, even on the assumption that £ the Pythagoreans ' are to be 
credited with the invention of eccentrics and epicycles, the difficul- 
ties are great, as Schiaparelli himself saw. 2 Who are the particular 
Pythagoreans who made the discovery? The problem which, 
according to Geminus, the Pythagoreans propounded of finding 
' how the phenomena could be accounted for by means of circular 
and uniform motions ' is almost identical with that which Sosigenes, 
on the authority of Eudemus, says that Plato set, ' What are the 
uniform and ordered movements by the assumption of which the 
facts about the movements of the planets can be accounted for ? '. 
If now the Pythagoreans had, by Plato's time, discovered the solu- 
tion by means of movable eccentrics and epicycles, Plato could not 
have been unaware of the fact, and he would not then have set the 
problem again in almost the same terms; Plato, however, makes 
no mention whatever of epicycles or eccentrics. Hence the Pytha- 
goreans in question could not have been the early Pythagoreans or 
any Pythagoreans up to the time of Philolaus (who was about half 
a century earlier than Plato) ; they must therefore be sought among 
the contemporaries of Plato or in the years immediately after his 
death ; indeed, if the hypothesis had been put forward in his life- 
time, we should have expected to find some allusion to it in his 
writings. We are, therefore, brought down to the period of Philip of 
Macedon and Alexander the Great. But it was in these reigns 
that the Pythagorean schools gradually died out, leaving the 

1 Hultsch, art. 'Astronomic' in Pauly-Wissowa's Real-Encyclopadie, § 14. 

2 Schiaparelli, Origine del sistema plane tario eliocentrico firesso i Greci, 
pp. 81-2. 



CH. χνπι HERACLIDES OF PONTUS 273 

name to certain fraternities whose objects were rather ascetic 
and religious than philosophical; 1 according to Diodorus the last 
Pythagorean philosophers lived about 366 B.C. 2 Schiaparelli is 
therefore obliged to assume that, ' if the schools ceased, their 
doctrines were not entirely lost/ and his whole case for crediting 
Heraclides or one of his contemporaries with the complete anticipa- 
tion of the system of Tycho Brahe really rests on this assumption 
combined with the statement of Diogenes Laertius that Heraclides 
' also heard the Pythagoreans \ 3 It is true that Schiaparelli has one 
other argument, which however seems to be an argument of despair. 
It is based on the passage, already quoted above (pp. 186-7), in which 
Aristotle, after speaking of the central fire of the 'so-called Pytha- 
goreans', says : 4 ' And no doubt many others, too, would agree (with 
the Pythagoreans) that the place in the centre should not be assigned to 
the earth, if they looked for the truth, not in the observed facts, but 
in a priori arguments. For they hold that it is appropriate to the 
worthiest object that it should be given the worthiest place. Now 
fire is worthier than earth . . .' Schiaparelli adds, ' On this passage 
Boeckh rightly observes that the reference is not to the past, but to 
opinions held in the time of Aristotle. The Pythagorean doctrines 
had ceased to be the object of teaching in special schools, but they 
survived in the opinions of many and in part found favour even in 
the Academy. From these reflections we draw the conclusion that 
the first idea of epicycles and of eccentrics was conceived towards 
the time of Philip or of Alexander, not among the pure Academics, 
nor in the Lyceum, but among those more independent thinkers 
who, like Heraclides, without forming a separate school, had 
remained faithful, at least so far as regards natural philosophy, to 
Pythagorean ideas, and for that reason could still with some truth 
be called Pythagoreans, especially by writers of a much later date.' 
That is to say, Nicomachus must, when claiming the discovery of 
eccentrics for the Pythagoreans, have been referring to certain 
persons whom Aristotle expressly distinguishes from that school, 
his ground for claiming those persons as Pythagoreans being that 
they were imbued with Pythagorean doctrines. It seems to me 

1 Zeller, i 5 , pp. 338-42, iii. 2 s , pp. 79 sqq. 

2 Diodorus, xv. 76 ; Zeller, r 5 , p. 339, note 2. s Diog. L. v. 86. 
4 Aristotle, De caelo ii. 13, 293 a 27-32. 

1410 Τ 



274 HERACLIDES OF PONTUS parti 

that, by this desperate suggestion, Schiaparelli practically gives 
away his case so far as it is based on Nicomachus. But, even if 
we assume Nicomachus to have been referring to the independent 
persons who, according to Aristotle, agreed in the theory of a 
central fire, this does not help Schiaparelli's argument, because in 
Aristotle's account of those persons' views there is no hint what- 
ever of eccentrics or epicycles. 

It is no doubt possible that Heraclides or one of his contem- 
poraries may, in the manner suggested, have arrived at the Tychonic 
system ; but I think that Schiaparelli has failed to establish 
this, and the probabilities seem to me to be decidedly against it. 
I judge mainly by the passage of Ptolemy (xii. i) about the two 
propositions proved by Apollonius and other geometers. Apol- 
lonius was born, probably, 125 years later than Heraclides. Now 
Heraclides certainly originated a particular hypothesis of epicycles, 
namely epicycles described by Venus and Mercury about the 
material sun as centre. By Apollonius's time the hypothesis of 
epicycles had become quite general, and such a generalization 
might easily come about in a period of a century and more. But 
the hypothesis of eccentrics had, by Apollonius's time, advanced only 
a very short way indeed towards a corresponding generality. Started 
to explain the movements of the superior planets, the hypothesis 
originally made the material sun the centre of the eccentric circle, 
and by Apollonius's time it had been only so far generalized as to 
allow the sun to be anywhere on the line joining the centre of the 
earth to the moving centre of the eccentric circle. This represents 
very little progress for a hundred years ; and the fact suggests that 
nothing like a hundred years had passed since the first formulation 
of the hypothesis in its most simple form, corresponding to the first 
form of the epicycle hypothesis. In other words, the Tychonic 
system was most probably completed by some one intermediate 
between Heraclides and Apollonius and nearer to Apollonius than 
Heraclides, if it was not actually reserved for Apollonius himself. 
And that there is a fair probability in favour of attributing the step 
to Apollonius himself seems to me to follow from two considera- 
tions. It is a priori less likely that the ' great geometer ' should 
merely have proved two geometrical propositions to show the effect 
of two hypotheses formulated by some of his predecessors, than that 



ch. χνιπ HERACLIDES OF PONTUS 275 

he should have attached the propositions to hypotheses, or to a 
comparison of hypotheses, which he was himself the first to develop ; 
and the fact that he takes the trouble to mention that the eccentric 
hypothesis only applies to the case of the three superior planets is 
more intelligible on the assumption that the hypothesis was at the 
time a new one, than it would be if the hypothesis had been familiar 
to mathematicians for some time. 

We have lastly to deal with a still greater claim put forward by 
Schiaparelli on behalf of Heraclides ; this is nothing less than the 
claim that it was Heraclides, and not Aristarchus of Samos, who 
first stated as a possibility the Copernican hypothesis. Schia- 
parelli's argument rests entirely on one passage, a sentence forming 
part of a quotation from Geminus which Simplicius copied from 
Alexander and embodied in his commentary on the Physics of 
Aristotle ;* and, inasmuch as this passage, as it stands in the MSS., 
is not only unconfirmed by any other passage in Greek writers, but 
is in direct conflict with other passages found in Simplicius himself, 
it calls for the very closest examination. As the context is itself 
important, I shall give a translation of the whole quotation from 
Geminus according to the text of Diels ; I shall then discuss the 
text and the interpretation of the particular sentence relied upon 
by Schiaparelli. The passage then of Simplicius is as follows : 

1 Alexander carefully quotes a certain explanation by Geminus 
taken from his summary of the Meteorologica of Posidonius. 
Geminus's comment, which is inspired by the views of Aristotle, 
is as follows : 

' " It is the business of physical inquiry to consider the substance 
of the heaven and the stars, their force and quality, their coming into 
being and their destruction, nay, it is in a position even to prove 
the facts about their size, shape, and arrangement ; astronomy, on 
the other hand, does not attempt to speak of anything of this kind, 
but proves the arrangement of the heavenly bodies by considera- 
tions based on the view that the heaven is a real κόσμος, and 
further, it tells us of the shapes and sizes and distances of the earth, 
sun, and moon, and of eclipses and conjunctions of the stars, as well 
as of the quality and extent of their movements. Accordingly, as 
it is connected with the investigation of quantity, size, and quality of 
form or shape, it naturally stood in need, in this way, of arithmetic 

1 Simplicius, in Phys. (ii. 2, 193 b 23), pp. 291. 21-292. 31, ed. Diels (1882). 

Τ 2 



ιηβ HERACLIDES OF PONTUS parti 

and geometry. The things, then, of which alone astronomy claims 
to give an account it is able to establish by means of arithmetic and 
geometry. Now in many cases the astronomer and the physicist 
will propose to prove the same point, e. g., that the sun is of great 
size or that the earth is spherical, but they will not proceed by the 
same road. The physicist will prove each fact by considerations of 
essence or substance, of force, of its being better that things should 
be as they are, or of coming into being and change ; the astronomer 
will prove them by the properties of figures or magnitudes, or by 
the amount of movement and the time that is appropriate to it. 
Again, the physicist will in many cases reach the cause by looking 
to creative force ; but the astronomer, when he proves facts from 
external conditions, is not qualified to judge of the cause, as when, for 
instance, he declares the earth or the stars to be spherical ; some- 
times he does not even desire to ascertain the cause, as when he 
discourses about an eclipse ; at other times he invents by way 
of hypothesis, and states certain expedients by the assumption of 
which the phenomena will be saved. For example, why do the 
sun, the moon, and the planets appear to move irregularly ? We 
may answer that, if we assume that their orbits are eccentric circles 
or that the stars describe an epicycle, their apparent irregularity 
will be saved ; and it will be necessary to go further and examine 
in how many different ways it is possible for these phenomena to be 
brought about, so that we may bring our theory concerning the 
planets into agreement with that explanation of the causes which 
follows an admissible method. Hence we actually find a certain 
person, Heraclides of Pontus, coming forward and saying that, even 
on the assumption that the earth moves in a certain way, while the 
sun is in a certain way at rest, the apparent irregularity with reference 
to the sun can be saved. For it is no part of the business of an astro- 
nomer to know what is by nature suited to a position of rest, and 
what sort of bodies are apt to move, but he introduces hypotheses 
under which some bodies remain fixed, while others move, and then 
considers to which hypotheses the phenomena actually observed 
in the heaven will correspond. But he must go to the physicist for 
his first principles, namely that the movements of the stars are simple, 
uniform and ordered, and by means of these principles he will then 
prove that the rhythmic motion of all alike is in circles, some being 
turned in parallel circles, others in oblique circles." Such is the 
account given by Geminus, or Posidonius in Geminus, of the dis- 
tinction between physics and astronomy, wherein the commentator 
is inspired by the views of Aristotle.' 

The important sentence for our purpose is that which I have 
italicized, and the above translation of it is a literal rendering of the 



CH. χνιπ HERACLIDES OF PONTUS 277 

reading of the MSS. and of Diels (81b και παρελθών τίς φησιν 
'Ηρακλείδης ό Ποντικός, οτι και κινούμενης πως της γης κ.τ.έ.). 
The reading and possible emendations of it will have to be discussed, 
but it will be convenient first of all to dispose of a question arising on 
the interpretation of the context. What is meant by ' the apparent 
irregularity with reference to the sun (ή περί τον ήλων φαινόμενη 
ανωμαλία)' ? Can this be so interpreted as to make it possible to 
take the motion of the earth to be rotation about its axis and not a 
motion of translation at all? Boeckh 1 took the πω? ('in a certain 
way ') used of the sun's remaining at rest (as it is also used of the 
earth's motion) to signify that the sun is not quite at rest ; and he 
thought that Heraclides meant that the sun and the heaven were 
only at rest so far as the general daily rotation was concerned, while 
the earth rotated on its own axis from west to east in 24 hours, but 
that the sun still performed its yearly revolution in the zodiac 
circle. This, however, does not account for the ' apparent irregu- 
larity or anomaly with reference to the sun ', which expression could 
not possibly be applied to the daily rotation. 

Martin 2 and Bergk 3 took the irregularity to be the irregularity 
of the sun's own motion in the ecliptic, by virtue of which the sun 
seems to go quicker at one time than at another, and the four 
seasons differ in length. But if 3 as Bergk apparently supposed, the 
two hypotheses which are contrasted are (1) the sun moving irregu- 
larly as it does and the earth completely devoid of any motion of 
translation, (2) the sun completely at rest and the earth with an 
irregular motion of translation, it is, as Schiaparelli says, impossible 
to get any plausible sense out of the passage. For the problem of 
explaining the irregularity of the sun's motion presents precisely the 
same difficulties on the one hypothesis as it does on the other; 
the substitution of one hypothesis for the other does not advance 
the question in any way, and it explains nothing. Martin saw this, 
and tried another explanation based on the use of the word πως, 
i in a certain way '. The mean speed of the sun, says Martin, is one 
thing, its anomaly is another ; the former is accounted for by the 

1 Boeckh, Das kosmische System des Platon, pp. 135-40. 

2 Martin in Memoires de V Academie des Inscriptions et Belles-Lettres, xxx. 
2 Θ partie, 1883, pp. 26 sqq. 

3 Bergk, Fiinf Abhandlungen zur Gesch. der griechischen Philosophic und 
Astronomie, Leipzig, 1883, p. 151. 



278 HERACLIDES OF PONTUS parti 

annual revolution ; the latter had to be otherwise accounted for, and 
one way of accounting for it was that of Callippus, who gave the 
sun two spheres more than Eudoxus assigned to it. Take now 
from the sun the small movement (of irregularity) only, thus leaving 
it at rest only ' in a certain sense ', and give the earth a small 
annual movement sufficient to explain the apparent anomaly of the 
sun. A mere rotation of the earth on its axis would not suffice ; 
the movement must be one of translation in the circumference of a 
circle, the result of which would be that, for the inhabitants of the 
earth, the solar anomaly would be the effect of a parallax, not 
daily, but annual, and dependent on the radius of the circle 
described by the earth in a year. That is, the earth must be supposed 
to accomplish, on the circumference of a small orbit round the centre 
of the universe, an annual revolution at a uniform speed from east 
to west, while the sun accomplishes from west to east its annual 
revolution about the same centre in a great orbit enveloping that of 
the earth. Schiaparelli shows the impossibility of this explanation. 
If we take from the sun only the small irregularity of its movement 
and leave it its mean movement in an enormous circle round the 
earth, how could any one properly describe this as making the sun 
1 stationary in a certain sense \ when at the same time the earth, 
which is made to describe a small orbit, is said to ' move in a certain 
sense ' ? Moreover, it is inadmissible to suppose that, in Heraclides' 
time, any one could have assumed that the place in the centre of 
the universe was occupied by nothings and that both the sun and 
the earth revolved about an ideal point ; the conception of revolu- 
tion about an immaterial point appeared later, in the generalized 
theory of epicycles and eccentrics, and we find no mention of it 
before Apollonius. 

But indeed there is nothing to suggest that Heraclides was aware 
of the small irregularities of the sun's motion, and it is therefore 
necessary to find another meaning for the expression * the apparent 
irregularity with reference to the sun ' (17 περί τον ήλων φαινόμενη 
ανωμαλία). I agree with Schiaparelli's view that it must be the 
same thing as Hipparchus and Ptolemy in the Syntaxis commonly 
describe as ' the irregularity relatively to the sun ' (17 προ? τον ήλων 
ανωμαλία or 17 τταρα. τον ήλων ανωμαλία), that is to say, that great 
inequality in the apparent movements of the planets, which alone 



ch. χνπι HERACLIDES OF PONTUS 279 

was known in the time of Heraclides and which manifests itself 
principally in the stationary points and retrogradations. It is true 
that in the particular sentence the planets are not mentioned, but 
they are mentioned in the sentence of Geminus which immediately 
precedes it, and, if our sentence is a quotation of words used by 
Heraclides, they no doubt followed upon a similar reference to the 
planets by Heraclides. 

If then the text as above translated is right, there is no escape from 
the conclusion that Heraclides actually put forward the Copernican 
hypothesis as a possible means of ' saving the phenomena \ But 
it is precisely the text of the sentence referring to Heraclides that 
gives rise to the greatest difficulties. The reading of the MSS. 
followed in the translation above is Sib- και παρελθών τις φησιν 
Ηρακλείδης ό Ποντικός, δτι και κινούμενης πόας της γης κ. τ. ε. Diels * 
is satisfied with this reading, which, he thinks, renders unnecessary 
the many scruples felt by scholars, and the emendation proposed 
by Bergk. 2 Gomperz, 3 on the other hand, says that after the most 
careful consideration he finds himself compelled to dissent from 
Diels' view of the passage. Schiaparelli observes that it is really 
impossible to suppose that a historian of sciences such as Geminus 
could have used the word τις, and said • a certain Heraclides of 
Pontus ', in speaking of a philosopher who was celebrated through- 
out antiquity and whom Cicero, a contemporary of Geminus, read 
and spoke of with great respect. This consideration may have 
been one of those which induced the editor of the Aldine edition 
to insert the word ελεγεν before δτι, a reading which involves the 
punctuation of the passage thus : διο και παρελθών τις, φησιν Ηρα- 
κλείδης ό Ποντικός, ελεγεν δτι. I think there is no doubt that 
Boeckh is right in his interpretation of παρελθών as ' having come 
forward ', which he supports by quoting a number of passages con- 
taining the same use of the word. According, therefore, to the 
reading of the Aldine edition we have a quotation from one of 
Heraclides' dialogues introduced by the parenthetical words in 
oratio recta, ' says Heraclides of Pontus,' and the translation will 

1 Diels, 'Uber das physikalische System des Straton' in Berliner Sitzungs- 
Berichte, 1893, p. 18, note I. 

2 Bergk, op. cit., p. 150. 

3 Gomperz, Griechische Denker, i 3 , p. 432. 



2 8o HERACLIDES OF PONTUS parti 

be, ' This explains too why u some one came forward ", as Heraclides 
of Pontus says, " and said that ....'" Bergk objects that, while 
παρελθών, * coming forward,' is used of one who comes forward in 
a public assembly, it is not, so far as he can find, used of the 
interlocutors in a dialogue. 1 This is, however, not conclusive, as 
such expressions marking the interposition of a new speaker may 
have been common in Heraclides' dialogues ; indeed we gather that 
there was a great deal of action in them. 2 A more substantial 
objection is the form of the quotation, the plunging direct, after 
the words Sib και, ' For which reason also,' into the actual words 
of Heraclides ' some one came forward and said '. This is, it must 
be admitted, extremely abrupt and awkward ; if Geminus had 
been quoting in this way, it would have been more natural to put 
the sentence in a different form, such as ' This is the reason too 
why, in a dialogue of Heraclides of Pontus, some one came forward 
and said that . . .'. 

Bergk's own suggestion for emendation is to omit the τι?, alter 
παρελθών into προβλθών, and write the sentence thus, Sib και προελ- 
θών φησιν 'Ηρακλείδη? ό Ποντικό? ελεγεν. ' For this reason too, 
he goes on to say w Heraclides of Pontus said that . . . " ' The 
words δώ και προελθών φησιν would thus be the words, not of 
Geminus, from whom the whole passage is quoted, but of 
Alexander, who is quoting ; these words would therefore come 
in between one textual citation from Geminus and another. I think 
this reading has nothing to commend it ; the omission of τι? is an 
objection to it, and the net result is a perfectly unnecessary 
interposition by Alexander, which moreover spoils the sense; 
' this is the reason, too, why Geminus goes on to say that Heraclides 
declared . . .' is not so good as ' this is the reason, too, why some 
one, according to Heraclides of Pontus, said . . .' 

1 omit a number of suggestions for replacing παρελθών by some 
other word or words ; they have no authority and, so long as 
Heraclides of Pontus remains in the sentence either as having 
himself held the view in question, or as having attributed it to some 
one else unnamed, they do not really affect the issue. 

I now come to Tannery's view of the passage, which is not only 

1 Bergk, op. cit., p. 149. 2 Otto Voss, Heraclides, p. 27. 



ch. χνπι HERACLIDES OF PONTUS 281 

that suggested by the ordinary principles of textual criticism, but 
furnishes a solution of the puzzle so simple and natural that it 
should, as it seems to me, carry conviction to the mind of any 
unbiased person. 1 As Tannery says, from the moment when it is 
realized that the insertion of the word ελεγεν does not, after all is 
said, suffice to remove all difficulties, we are thrown back upon 
the text of the MSS. as established by Diels, δώ και παρελθών τις 
φησιν 'Ηρακλείδης ό Ποντικός, οτι καϊ κινούμενης πως της γης κ.τ.ε., 
and we have to consider in what way an error could have crept 
into the text. Now it ' leaps to the eyes ' that, if the original text 
said simply δώ καϊ παρελθών τίς φησιν οτι και κινούμενης πως της 
γης κ.τ.ε., it was the easiest thing in the world for a glossarist to 
insert in the margin, in explanation of τις, the name of Heraclides 
of Pontus, which would then naturally find its way into the text. 
If the name is left out, everything is in perfect order. The passage 
in its context is then as follows : ' Why do the sun, moon, and 
planets appear to move irregularly ? We may answer that, if we 
assume that their orbits are eccentric circles or that the stars 
describe an epicycle, their apparent irregularity will be saved ; and 
it will be necessary to go further and examine in how many different 
ways it is possible for these phenomena to be brought about, 
so that we may bring our theory concerning the planets into 
agreement with that explanation of the causes which follows an 
admissible method. This is why one astronomer has actually 
suggested that, by assuming the earth to move in a certain way 
and the sun to be in a certain way at rest . . .' Nothing clearer 
or more correct could possibly be desired ; the different hypotheses 
would then all alike be stated in general terms without the names 
of their authors, whereas nothing could be more awkward than 
that Geminus, after speaking of the hypotheses of eccentrics and 
epicycles in this way, should change the form of statement and 
bring in quite abruptly an historical fact about one particular person 
by name, or a textual citation from a work of his. Even Gomperz 2 
admits that it is possible that Ηρακλείδης 6 Ποντικός may have 
been inserted ' by a (well-informed) reader ' ; but the ' well-informed ' 

1 Tannery, ' Sur Heraclide du Pont,' in Revue des Etudes grecques, xii, 1 899, 
pp. 305-11. 

2 Gomperz, loc. cit. 



28α HERACLIDES OF PCNTUS parti 

is a pure assumption on his part, due, I think, merely to bias, for 
how can we possibly pronounce the reader to have been ' well- 
informed ' when there is absolutely no other evidence telling in his 
favour ? 

If then, as seems to me inevitable, the words Ήρακλβίδης ό 
Ποντικός are rejected as an interpolation, the view of Schiaparelli 
based upon the passage must be given up, and there remains no 
ground for disputing the accuracy of the other definite statement 
by Aetius to the effect that ' Heraclides of Pontus and Ecphantus 
the Pythagorean make the earth move, yet not in the sense of 
translation but with a movement of rotation ', 1 confirmed as it is 
by the sharp distinction drawn in one place by Simplicius between 
those who supposed the earth to have a motion of translation and 
Heraclides who supposed it to rotate about its axis. 2 

If it is asked whom Geminus had in mind when using the 
expression τίς φησιν, we can have no hesitation in answering that 
it was Aristarchus of Samos, for it is to him that all ancient 
authorities agree in attributing the suggestion of the heliocentric 
system. 

It is not possible to say at what time the interpolation of the 
name of Heraclides into the passage took place. There is nothing 
to <show, for instance, whether it was made in the archetype of the 
MSS. of Simplicius or in the sources from which he drew. As he 
did not quote Geminus directly, but copied the quotation of 
Alexander, it is a question whether the gloss is earlier or later than 
Alexander, or even due to Alexander himself. I agree with Tannery 
that an annotator of the second or third century of our era, at 
a period when Heraclides was sufficiently well known through the 
Doxographi as having attributed a movement to the earth, might 
very well, on reading the first words, κινούμενης πω? τή? γή$ (' if 
the earth moves in a certain way '), have immediately thought of 
Heraclides rather than Aristarchus, and have written the name 
of the former in the margin without looking forward to see what 
were the words immediately following these. ' In any case,' Tannery 
concludes, ' the attribution to Heraclides Ponticus of the heliocentric 

1 Aet. iii. 13. 3 (D. G. p. 378 ; Vors. i 2 , p. 266. 5-7). 

2 Simplicius on De caelo ii. 14 (297 a 2), p. 541. 27-9, Heib. See above, 
p. 255. 



CH.xviir - HERACLIDES OF PONTUS 283 

system does not in any way rest on the authority of Posidonius 
or of Geminus ; it is the act of an anonymous annotator of uncertain 
date, and probably the result of a simple inadvertence only too 
easy to commit ; it must therefore be considered as null and void.' 

It may be added that it is hardly a priori surprising that such 
extensive claims on behalf of Heraclides should prove, on examina- 
tion, to be in part unsustainable. It was much to discover, as he 
did, the rotation of the earth about its axis and the fact that Venus 
and Mercury revolve round the sun like satellites ; it would seem 
a priori almost incredible that the complete Tychonic system should 
have been evolved in Heraclides' lifetime and ' perhaps ' by 
Heraclides himself, and that he should also have suggested the 
Copernican hypothesis. 



XIX 

GREEK MONTHS, YEARS, AND CYCLES 1 

Although there is controversy as to whether in the earliest 
times (e. g. with Homer and Hesiod) the day was supposed to 
begin with the morning or evening, it may be taken as established 
that in historic times the day, for the purpose of the calendar, began 
with the evening, both at Athens and in Greece generally. As 
regards Athens the fact is stated by Gellius on the authority of 
Varro, who, in describing the usage of different nations in this 
respect, said that the Athenians reckoned as one day the whole 
period from one sunset to the next sunset ; 2 the testimony of Pliny 3 
and Censorinus 4 is to the same effect. The practice of regarding 
the day as beginning with the evening is natural with a system of 
reckoning time by the moon's appearances ; for a day would 
naturally be supposed to begin with the time at which the light of 
the new moon first became visible, i.e. at evening. 

There is no doubt that, from the earliest times, the Greek month 
(μήν) was lunar, that is, a month based on the moon's apparent 
motion. But from the first there began to be felt, among the 
Greeks as among most civilized peoples, a desire to bring the 
reckoning of time by the moon into correspondence with the 
seasons of the year, for the sake of regulating the times of sacri- 
fices to the gods which had to be offered at certain periods in the 
year ; hence there was from the beginning a motive for striving 
after the settlement of a luni-solar year. The luni-solar year thus 
had a religious origin. This is attested by Geminus, who says : 5 

' The ancients had before them the problem of reckoning the 
months by the moon, but the years by the sun. For the legal and 

1 For the contents of this chapter I am almost entirely indebted to the 
exhaustive work of F. K. Ginzel, Handbuch der mathematischen und technischen 
CJironologie, vol. ii of which appeared in the nick of time (191 1). 

2 Gellius, Nod. Att. iii. 2. 2. 3 Pliny, N. H. ii. c. 77, § 188. 

4 Censorinus, De die natali, c. 23. 3. 

5 Geminus, Isagoge, c. 8, 6-9, p. 102. 8-26, Manitius. 



GREEK MONTHS, YEARS, AND CYCLES 285 

oracular prescription that sacrifices should be offered after the 
manner of their forefathers was interpreted by all Greeks as mean- 
ing that they should keep the years in agreement with the sun and 
the days and months with the moon. Now reckoning the years 
according to the sun means performing the same sacrifices to the 
gods at the same seasons in the year, that is to say, performing 
the spring sacrifice always in the spring, the summer sacrifice 
in the summer, and similarly offering the same sacrifices from year 
to year at the other definite periods of the year when they fell due. 
For they apprehended that this was welcome and pleasing to the 
gods. The object in view, then, could not be secured in any other 
way than by contriving that the solstices and the equinoxes should 
occur in the same months from year to year. Reckoning the days 
according to the moon means contriving that the names of the 
days of the month shall follow the phases of the moon.' 

At first the month would be simply regarded as lasting from 
the first appearance of the thin crescent at any new moon till the 
next similar first appearance. From this would gradually be 
evolved a notion of the length of a moon-year. A rough moon-year 
would be 1 2 moon-months averaging 29-I days ; but it was necessary 
that a month should contain an exact number of days, and it was 
therefore natural to take the months as having alternately 29 and 30 
days. These ' hollow ' and ' full ' months are commonly supposed 
to have been introduced at Athens by Solon (who was archon in 
594/3 B.C.), since he is said to have 'taught the Athenians to 
reckon days by the moon ', 1 But it can hardly be doubted that 
1 full ' and ' hollow ' months were in use before Solon's time ; 
Ginzel therefore thinks that Solon's reform was something different. 
We shall revert to this point later. 

At the same time, alongside the 'full' and ( hollow' months of the 
calendar, popular parlance invented a month of 30 days, as being 
convenient to reckon with. Hippocrates makes 280 days = 9 
months 10 days; 2 Aristotle speaks of 72 days as i/5th of a year ; 3 
the riddle of Cleobulus implies 12 months of 30 days each; 4 the 
original division of the Athenian people into 4 φυλαί, 12 φρατρίαι,, 
and 360 ykvi) is explained by Philochorus as corresponding to the 
seasons, months, and days of the year. 5 In the Courts a month 

1 Diog. L. i. 59. 2 Hippocrates, De carnibus, p. 254. 

8 Aristotle, Hist. an. vi. 20, 574 a 26. 4 Diog. L. i. 91. 

5 Suidas, S. V. γεννήται. 



286 GREEK MONTHS, YEARS, AND CYCLES parti 

was reckoned at 30 days, and wages were reckoned on this basis, 
e.g. daily pay of 2 drachmae makes for 13 months 780 drachmae 
(2 X 30 χ 13). 1 From such indications as these it has been inferred 
that the Greeks had at one time years of 360 days and 390 days 
respectively. Indeed, Geminus says that ' the ancients made the 
months 30 days each, and added the intercalary months in alter- 
nate years (παρ' kviavrov) \ 2 Censorinus has a similar remark ; when, 
he says, the ancient city-states in Greece noticed that, while the sun 
in its annual course is describing its circle, the new moon sometimes 
rises thirteen times, and that this often happens in alternate years, 
they inferred that I2§ months corresponded to the natural year, and 
they therefore fixed their civil years in such a way that they made 
years of 12 months and years of 13 months alternate, calling each 
of such years ' annus vertens ' and both years together a great year. 3 
Again, Herodotus 4 represents Solon as saying that the 70 years of 
a man's life mean 25,200 days, without reckoning intercalary 
months, but, if alternate years are lengthened by a month, there 
are 35 of these extra months in 70 years, making 1,050 days more 
and increasing the total number of days to 26,250. But under this 
system the two-years period (called nevertheless, according to 
Censorinus, trieteris because the intercalation took place ' every 
third year ') would be more than 7 days too long in comparison 
with the sun, and in 20 years the calendar would be about i\ months 
wrong in relation to the seasons. This divergence is so glaring that 
Ginzel concludes that the system cannot have existed in practice. 
He suggests, in explanation of Geminus's remark, that Geminus is 
not to be taken literally, but is in this case merely using popular 
language (cf. his remark that 90 days = 3 months 5 ) ; he regards 
Censorinus's story as suspicious because in the following sentence 
Censorinus says that the next change was to a pentaeteris of four 
years each, which involves the supposition that the Greeks of, say, 
the eighth or ninth century B.C., had already anticipated the Julian 
system ; moreover, Geminus says nothing of a four-years period at 
all (whether called tetraeteris or pentaeteris) but passes directly to 
the octa'eteris which, according to him, was the first period that the 
ancients constructed. 

1 Corp. Inscr. Att. ii. 2, no. 834 c, 1. 60 (p. 532). 2 Geminus, Isagoge, c. 8. 26. 
8 Censorinus, De die natali, c. 18. 2. * Herodotus, i. 32. 

5 Geminus, Isagoge, c. 8. 30, p. 112. 7, 10. 



ch.xix GREEK MONTHS, YEARS, AND CYCLES 287 

On the alternation of ■ full ' and ' hollow ' months an apparently- 
interpolated passage in Geminus says ι 1 

'The moon-year has 354 days. Consequently they took the 
lunar month to be 29J days and the double month to be 59 days. 
Hence it is that they have hollow and full months alternately, 
namely because the two-months period according to the moon is 
59 days. Therefore there are in the year six full and six hollow 
months. This then is the reason why they make the months full 
and hollow alternately.' 

The octaeteris. 

Geminus's account of the eight-years cycle follows directly on 

what he says of the supposed ancient system of alternating years 

of 12 and 13 months of 30 days each. 

' Observation having speedily proved this procedure to be incon- 
sistent with the true facts, inasmuch as the days and the months 
did not agree with the moon nor the years keep pace with the sun, 
they sought for a period which should, as regards the years, agree 
with the sun, and, as regards the months and the days, with the 
moon, and should contain a whole number of months, a whole 
number of days, and a whole number of years. The first period 
they constructed was the period of the octaeteris (or eight years) 
which contains 99 months, of which three are intercalary, 2922 days, 
and 8 years. And they constructed it in this way. Since the year 
according to the sun has 365^ days, and the year according to the 
moon 354 days, they took the excess by which the year according 
to the sun exceeds the year according to the moon. This is 11J 
days. If then we reckon the months in the year according to the 
moon, we shall fall behind by ii\ days in comparison with the solar 
year. They inquired therefore how many times this number of 
days must be multiplied in order to complete a whole number 
of days and a whole number of months. Now the number [nj] 
multiplied by 8 makes 90 days, that is, three months. Since then 
we fall behind by \\\ days in the year in comparison with the sun, 
it is manifest that in 8 years we shall fall behind by 90 days, that 
is, by 3 months, in comparison with the sun. Accordingly, in each 
period of 8 years, three intercalary (εμβόλιμοι) months are reckoned, 
in order that the deficiency which arises in each year in comparison 
with the sun may be made good, and so, when 8 years have passed 
from the beginning of the period, the festivals are again brought 
into accord with the seasons in the year. When this system is 
followed, the sacrifices will always be offered to the gods at the 
same seasons of the year. 

1 Geminus, Isagoge, c. 8. 34-5, pp. 1 12. 28- 114. 7. 



288 GREEK MONTHS, YEARS, AND CYCLES parti 

' They now disposed the intercalary months in such a way as to 
spread them as nearly as possible evenly. For we must not wait 
until the divergence from the observed phenomena amounts to 
a whole month, nor yet must we get a whole month ahead of the 
sun's course. Accordingly they decided to introduce the inter- 
calary months in the third, fifth, and eighth years, so that two of 
the said months were in years following two ordinary years, and 
only one followed after an interval of one year. 1 But it is a matter 
of indifference if, while preserving the same disposition of (i.e. inter- 
vals between) the intercalary months, you put them in other years.' 2 

Here then we have an account which purports to show how the 
octaeteris was first arrived at, the supposition being that it was 
based on a solar year of 365^ days. Ginzel, however, thinks it 
impossible that this can have been the real method, because the 
evaluation of the solar year at 365^ days could hardly have been 
known to the Greeks of, say, the 9th and 8th centuries B.C. ; this, 
he thinks, is proved by the erroneous estimates of the length of the 
solar year which continued to be put forward much later. 

Ginzel considers that the octaeteris was first evolved as the result 
of observation of the moon's motion, which was of course easier to 
approximate to within a reasonable time. The alternation of 6 full 
with 6 hollow months gives a moon-year of 354 days ; but the true 
moon-year exceeds this by 0-36707 day, and hence, after about 2-§ 
moon-years, a day would have to be added in order to keep the 
months in harmony with the phases ; that is to say, at such inter- 
vals, there would have to be a year of 3$$ days. Now this rate 
of intercalation corresponds nearly to the addition of 3 days in 
a period of 8 moon-years, i. e. to a cycle of 8 moon-years in which 
5 have 354, and 3 have 355 days, each. (And, as a matter of fact, 
the same proportion of 5 : 3 serves very roughly to bring the moon- 
year into agreement with the solar year, for we have only to reckon, 
in a cycle of 8 solar years, 5 moon-years of 354 days and 3 of 
384 days.) 3 Ginzel cites evidence showing that particular years 
actually had 355 days and 384 days, e.g. 01. 88, 3 = 355 days, 
Ol. 88, 4 = 354 days, 01. 89, 1 = 384 days, and 01. 89, 2 = 355 

δι ην αΐτ'ιαν τους εμβόλιμους μήνας έταζαν αγεσθαι iv τω τρ'ιτω era και πέμπτω 
και ογδόω, δυο μεν μήνας μεταξύ δύο ετών πιπτόντων, ενα δε μεταξύ ενιαυτοΰ ενός 
αγομένου. 

2 Geminus, Isagoge 8. 26-335 ΡΡ• no. 14- 112. 27. s Ginzel, ii. 330-1. 



ch.xix THE OCTAETERIS 289 

days. 1 The method by which the octaeteris was evolved is, he thinks, 
something of this sort. Having from observation of the moon con- 
structed an 8-years period containing 5 moon-years of 354 days and 
3 intercalated years of 355 days each, making a total of 3,835 days, 
the Greeks, by further continual observation directed to fixing the 
duration of the phases exactly, would at last come to notice that, 
after 8 returns of the sun to the same azimuth-point on the horizon, 
the phases fell nearly on the same days once more, and also that the 
sun returned to the same azimuth-point for the eighth time after about 
99 lunar months. Now, if the ancients had divided the 2,835 days of 
8 moon-years by 96, they would have found the average lunar month 
to contain 29-U days ; and again, if they had multiplied this by 99, 
they would have obtained 2,923•!! or nearly 2,9 23^ days. But the 
first inventors of the octaeteris certainly did not make the 8 solar years 
contain 2,923! days ; this, we are told, was a later improvement on 
the 2,922 days which, according to Geminus, the first octaeteris con- 
tained. No doubt the first discoverers of it would notice that 99 
times 29-I days is 2,920^ days, that is to say, approximately 8 years 
of 365 days (= 2,920 days). This may have been what led them 
to construct a luni-solar octaeteris. But why did they give it 2,922 
days ? Ginzel suggests that, as the octaeteris was thus shown to be 
very useful for the purpose of bringing into harmony the motions 
of the sun and moon, the Greeks would be encouraged to try to 
obtain a more accurate estimate of the average length of the lunar 
month. If then, for example, they had assumed 29§§ days as the 
average length, they would have found, at the end of an octaeteris, 
that they were only wrong by 0-3 of a day relatively to the moon, 
but were nearly two days ahead in relation to the sun. 2 This 
might perhaps lead them to conjecture that the solar year was 
a little longer than 365 days ; and they may have hit upon 365% 
days by a sort of guess. This would give 29IJ days as the length 
of the lunar month. Ginzel thinks that the gradual process by 
which the Greeks arrived at the 2,922 days may have lasted from 
the 9th or 8th century into the 7th. 3 This, he suggests, may explain 

1 Ginzel, ii. 341-3. 

2 2 9M X 99 = 2 9 2 3•8 days (against 2923*528, the correct figure) ; 8 solar years 
have 8 χ 365-2422 = 2921*938 days. 

3 Ginzel, ii. 376, 377. 

1410 υ 



290 GREEK MONTHS, YEARS, AND CYCLES parti 

the fact that we find mentions or indications of eight-years periods 
going back as far as the mythical age. Thus Cadmus passed an 
'eternal' (άίδιος) year (i.e. says Ginzel, an 8-year year) in servitude 
for having slain the dragon of Ares ; similarly Apollo served 8 years 
with Admetus after he killed the dragon Python. The Daphne- 
phoria were celebrated every 8 years ; in the procession connected 
with the celebration an olive staff was carried with a sphere above 
(the sun), a smaller one below (the moon), and still smaller spheres 
representing other stars, while 365 purple bands or ribbons were 
also attached, representing the days of the solar year. The Pythian 
games were also, at the beginning, eight-yearly. Kingships were 
offices held for eight years (thus Minos spoke with Zeus, the great 
God, 'nine-yearly'). 1 According to Plutarch the heaven was 
observed at Sparta by the Ephors on a clear night once in eight 
years. 2 These cases, however, though showing that 8-years periods 
were recognized and used in various connexions, scarcely suffice, I 
think, to prove the existence in such very early times of an accu- 
rately measured period of 2,922 days. Ginzel, in arguing for so 
early a discovery of the octaeteris of 2,922 days, departs consider- 
ably from the views of earlier authorities on chronology. Boeckh 
thought that the octaeteris was introduced by Solon, and that the 
first such period actually began with the beginning of the year at 
the first new moon after the summer solstice in Ol. 46, 3, i. e. 7th 
July, 594 B. c. 3 As regards the period before Solon, Boeckh went, 
it is true, so far as to suggest that, as early as 642 B.C., there may 
have been a rough octaeteris in vogue which was not actually fixed 
or exactly observed ; this, however, was only a conjecture. Ideler 4 
argued that the octaeteris could not be as old as Solon's time 
(594/3 B.C.) or even as old as 01. 59 (544-540), because so accurate 
a conception is in too strong a contrast to what we know of the 
state of astronomical knowledge in Greece at that time. As regards 
Solon's reforms, we are told 5 that he prescribed that the day in the 

1 Odyssey xix. 178, 179. 2 Plutarch, Agis, c. II. 

3 The practice of beginning the year in the summer (with the month 
Hecatombaion) is proved by Boeckh to have existed during the whole of the 
fifth century. It was probably much older in Attica ; the transition (if the Attic 
year previously began in the winter) may have taken effect in the time of Solon. 

4 Ideler, Historische Untersuchungen iiber die astronomischen Beobachtungen 
der Alten, p. 191. 

6 Plutarch, Solon, c. 25. 



CH.xix THE OCTAETERIS 291 

course of which the actual conjunction at the new moon took place 
should be called k'vr) και pea, the ' last and new ' or ' old-new ', 
and that he called the following day νονμηνία (new moon), which 
therefore was the first day belonging wholly to the new month. 
Diogenes Laertius says that Solon taught the Athenians ' to 
reckon the days according to the moon ' ; * and Theodorus Gaza, 
a late writer, it is true, says that Solon ' ordered everything in 
connexion with the year generally better'. 2 Boeckh, as already 
stated, thought that Solon's reform consisted in the introduc- 
tion of the octaeteris. Ginzel, however, holding as he does that 
the octaeteris of 2,922 da'ys was discovered much earlier, considers 
that Solon's reform had to do with the improvement on this 
figure by which 99 lunations were found to amount to 2,923-^ 
days, a discovery which led to the formulation of the 16-years and 
1 60-years periods presently to be mentioned ;, this may be inferred, 
according to Ginzel, from the fact that the accounts show Solon's 
object to have been the bringing of the calendar specially into 
accordance with the moon. But it is difficult to accept Ginzel's 
view of the nature of Solon's reform in the face of another statement 
as to the authors of the octaeteris. Censorinus says : 

'This octaeteris is commonly attributed to Eudoxus, but others 
say that Cleostratus of Tenedos first framed it, and that it was 
modified afterwards by others who put forward their octaeterides 
with variations in the intercalations of the months, as did Harpalus, 
Nauteles, Menestratus, and others also, among whom is Dositheus, 
who is most generally identified with the octaeteris of Eudoxus.' 3 

Now we know nothing of the date of Cleostratus, except that 
he came after Anaximander ; for Pliny says that Anaximander is 
credited by tradition with having discovered the obliquity of the 
zodiac in 01. 58 (548-544 B. C.), after which {deinde) Cleostratus 
distinguished the signs in it. 4 Thus Cleostratus may have lived 
soon after 544 B. C. Ginzel seems to admit that Cleostratus was 
the actual founder (' eigentliche Begriinder ') of the octaeteris? Of 
Harpalus, who was later than Cleostratus but before Meton 
(432 B. c), we only know that he formed a period which brought 
the moon into agreement with the sun after the latter had revolved 

1 Diog. L. i. 59. 

2 Theodorus Gaza, c. 8 and 15. 3 Censorinus, De die natali, 18. 5. 
4 Pliny, Ν. H, ii. c. 8, § 31. 5 Ginzel, ii, p. 385. 

U 2 



292 GREEK MONTHS, YEARS, AND CYCLES parti 

'through nine winters', 1 which statement must, as Ideler says, be 
due to a misapprehension of the meaning of the words ' nono quoque 
anno'. According to Censorinus, Harpalus made the solar year 
consist of 365 days and 13 equinoctial hours. 2 Eudoxus's variation 
will be mentioned later. 

The 16-years and 160-years cycles. 

After describing the octaeteris of 2,922 days, Geminus proceeds 
thus: 

1 If now it had only been necessary for us to keep in agreement 
with the solar years, it would have sufficed to use the aforesaid 
period in order to be in agreement with the phenomena. But as we 
must not only reckon the years according to the sun, but also the 
days and months according to the moon, they considered how this 
also could be achieved. Thus the lunar month, accurately measured, 
having 29^ ^ days, while the octaeteris contains, with the inter- 
calary months, 99 months in all, they multiplied the 29J ^ days of 
the month by the 99 months ; the result is 2,923•! days. Therefore 
in eight solar years there should be reckoned 2,923-! days according 
to the moon. But the solar year has 365% days, and eight solar 
years contain 2922 days, this being the number of days obtained by 
multiplying by 8 the number of days in the year. Inasmuch then 
as we found the number of days according to the moon which are 
contained in the 8 years to be 2,923^, we shall, in each octaeteris, 
fall behind by i-J days in comparison with the moon. Therefore in 
16 years we shall be behind by 3 days in comparison with the 
moon. It follows that in each period of 16 years three days have 
to be added, having regard to the moon's motion, in order that we 
may reckon the years according to the sun, and the months 
and days according to the moon. But, when this correction is 
made, another error supervenes. For the three days according to 
the moon which are added in the 16 years give, in ten periods 
of 16 years, an excess (with reference to the sun) of 30 days, that is 
to say, a month. Consequently, at intervals of 160 years, one of 
the intercalary months is omitted from (one of) the octaeterides ; 
that is, instead of the three (intercalary) months which fall to be 
reckoned in the eight years, only two are actually introduced. 
Hence, when the month is thus eliminated, we start again in agree- 
ment with the moon as regards the months and days, and with the 
sun as regards the years.' ΰ 

1 Festus Avienus, Prognost. 41, quoted by Ideler, op. cit., p. 191. 

2 Censorinus, De die natali, 19. 2. 

8 Geminus, Jsagoge, 8. 36-41, pp. 114. 8- 116. 15. 



ch.xix METON'S CYCLE 293 

This passage explains itself; it is only necessary to add that 
there is no proof that the 16-years period was actually used. The 
] 60-years period was, however, presupposed in Eudoxus's octaeteris, 
the first of which, according to Boeckh, may have begun in 381 or 
373 B. C. (01. 99, 4 or 01. 101, 4) on 22/23 July, the * first day of 
Leo ', i. e. the day on which the sun entered the sign of Leo ; the 
effect was that, after 20 octaeterides and the dropping of 30 days, 
the beginning of the solar year was again on ' the first of Leo '. 
In Eudoxus's system, then, theluni-solar reckoning was independent 
of the solstices. 1 According to the Eudoxus-Papyrus (Ars Eudoxi) 
the intercalary months came in the 3rd, 6th, and 8th years of 
Eudoxus's octa'eteris. 

Meton's cycle. 

Curiously enough, Meton is not mentioned by Geminus as the 
author of the 19-years cycle; his connexion with it is, however, 
clearly established by other evidence. Diodorus has the following 
remark with regard to the year of the archonship of Apseudes 
(01. 86, 4 = 433A B • C). 

1 In Athens Meton, the son of Pausanias, and famous in astro- 
nomy, put forward the so-called 19-years period (ίννβακαιδβκα- 
ετηρίδα); he started (αρχήν ποιησάμζνος) from the 13th of the 
Athenian month Skirophorion.' 2 

Aelian says that Meton discovered the Great Year, and ' reckoned 
it at 19 years', 3 and also that 'the astronomer Meton erected 
pillars and noted on them the solstices '. Censorinus, too, says that 
Meton constructed a Great Year of 19 years, which was accordingly 
called enneadecaeterisl• Euctemon, whom Geminus does mention, 
assisted Meton in the matter of this cycle. 

Geminus's account of the cycle shall be quoted in full : 

' Accordingly, as the octaeteris was found to be in all respects 
incorrect, the astronomers Euctemon, Philippus, and Callippus [the 
phrase is οι περί Εύκτήμονα /ere., as usual] constructed another 
period, that of 19 years. For they found by observation that in 19 
years there were contained 6940 days and 235 months, including 
the intercalary months, of which, in the 19 years, there are 7. 
[According to this reckoning the year comes to have S^Ots days.] 

1 Boeckh, Ueber die vierjdhrigen Sonnenkreise der Alten, 1863, pp. 159-66. 

2 Diodorus Siculus, xii. 36. 3 Aelian, V. H. x. 7. 
4 Censorinus, De die natali, 18. 8. 



2 9 4 GREEK MONTHS, YEARS, AND CYCLES part ι 

And of the 235 months they made no hollow and 125 full, so that 
hollow and full months did not always follow one another alter- 
nately, but sometimes there would be two full months in succession. 
For the natural course of the phenomena in regard to the moon 
admits of this, whereas there was no such thing in the octaeteris. 
And they included no hollow months in the 235 months for the 
following reason. As there are 235 months in the 19 years, they 
began by assuming each of the months to have 30 days ; this gives 
7,050 days; Thus, when all the months are taken at 30 days, the 
7,050 days are in excess of the 6,940 days ; the difference is (no 
days), and accordingly they make no months hollow in order to 
complete, in the 235 months, the 6,940 days of the 19-years period. 
But, in order that the days to be eliminated might be distributed as 
evenly as possible, they divided the 6,940 days by no ; this gives 
63 days. 1 It is necessary therefore to eliminate the [one] day after 
intervals of 63 days in this cycle. Thus it is not always the 30th 
day of the month which is eliminated, but it is the day falling after 
each interval of 63 days which is called έξα,ίρίσιμος.' 2 

The figure of 365^ days = 365 days 6 h i8 m 56«9 8 , and is still 30 
minutes 11 seconds too long in comparison with the mean tropic 
year ; but the mean lunar month of Meton is 29 days i2 h 45 m 57-§ 8 , 
which differs from the true mean lunar month by not quite I minute 
54 seconds. When Diodorus says that, in putting forward his 19- 
years cycle, Meton started from the 13th of Skirophorion (which 
was the 13th of the last month of Apseudes' year = 27th June, 432), 
he does not mean that the first year of the period began on that 
date ; this would have been contrary to the established practice. 
The beginning of the first year (the 1st Hekatombaion of that year) 
would be the day of the first visibility of the new moon next after 
the summer solstice, i.e. in this case 16th July, 432. The 13th 
Skirophorion was the day of the solstice, and we have several 
allusions to Meton's observation of this ; 3 presumably, therefore, 

1 What should really have been done is to divide 7,050 by no; this would 
give 64 as quotient, and the result would be that every 64th day would have to 
be eliminated, i.e. the day following successive intervals of 63 days. This fact 
would easily cause 63 to be substituted for the quotient, and this would lead to 
6,940 being taken as the number to be divided by no. 

2 Geminus, Isagoge, c. 8. 50-6, pp. 120. 4-122. 7. 

3 Philochorus (Schol. ad Aristoph. Aims 997) says that, under Apseudes, 
Meton of Leuconoe erected a heliotropion near the wall of the Pnyx, and it was 
doubtless there that he observed the solstice. Ptolemy says of this observation 
that it was on the 21st of the Egyptian month Phamenoth in Apseudes' 
year (Syntaxis, iii. 2, vol. i, p. 205, Heib.). This is confirmed by the discovery 



CH.xix CALLIPPUS'S CYCLE 295 

Diodorus meant, not that the first year of Meton's cycle began on 
that day, but that it was on that day that Meton began his para- 
pegma (or calendar). 1 Ginzel 2 gives full details of the many divergent 
views as to the date from which Meton's cycle was actually intro- 
duced at Athens. Boeckh put it in 01. 112, 3 = 330/29 B. C, Unger 
between 01. 109, 3 (342/1 B. c.) and 01. in, 1 (336/5 Β. C.). Schmidt 
holds that Meton's cycle was introduced in 342 B.C., but in a 
modified form. The 235 months of the 19-years cycle contained, 
according to the true mean motion of the moon, 235 χ 29-53059 days, 
or 6,939 days and about ι6•| hours. Consequently after 4 cycles 
there was an excess of four times the difference between 6,940 days 
and 6,939 days i6-| hours, or an excess of 1 day 6 hours ; after 10 
cycles an excess of 3 days 3 hours, and so on. The Athenians, 
therefore, according to Schmidt, struck out one day in the 4th, 7th, 
10th, 13th, i6th, 20th, and 23rd cycles, making these cycles 6,939 
days each. But, as Ginzel points out, the confusions in the calendar 
which occurred subsequently tell against the supposition that such 
a principle as that assumed by Schmidt was steadily followed in 
Athens from 342 B. C. 

Callippus's cycle of 76 years. 

Geminus follows up his explanation with regard to the Metonic 
cycle thus ; 

* In this cycle [the Metonic] the months appear to be correctly 
taken, and the intercalary months to be distributed so as to secure 
agreement with the phenomena ; but the length of the year as 
taken is not in agreement with the phenomena. For the length of 
the year is admitted, on the basis of observations extending over 
many years, to contain 36$ days, whereas the year which is 
obtained from the 19-year period has 365-^9 days, which number of 

of a fragment of a parapegma at Miletus which alludes to the same observation 
of the summer solstice on 13th Skirophorion or 21st Phamenoth, and adds that 
in the year of . . . €vktos the solstice fell on 14th Skirophorion or the Egyptian 
nth Payni. Diels showed from another fragment that the archon must have 
been Polykleitos (110/109 B.C.), so that the second observation of the solstice 
mentioned in the fragment must have been on 27th June, 109, i.e. in the last 
(19th) year of the 17th Metonic cycle (Ginzel, ii, pp. 423, 424). 

1 The παράπηγμα was a posted record (παραπηγνυμι), a sort of almanac giving, 
for a series of years, the movements of the sun, the dates of the phases of the 
moon, the risings and settings of certain stars, besides ςπισημασίαι or weather 
indications. 

2 Ginzel, op. cit., ii. 418, 430, 431, 442 sqq. 



296 GREEK MONTHS, YEARS, AND CYCLES parti 

days exceeds $6$ by ygth of a day. On this ground Callippus and 
the astronomers of his school corrected this excess of a (fraction of 
a) day and constructed the 76-years period (έκκαίεβδομηκονταβτη- 
ρίδα) out of four periods of 19 years, which contain in all 940 
months, including 28 intercalary, and 27,759 days. They adopted 
the same arrangement of the intercalary months. And this period 
appears to agree the best of all with the observed phenomena.' l 

With Meton's year of 365 J ^ Έ days (6,940 divided by 19), four 
periods of 19 years amount of course to 27,760 days, and the effect 
of Callippus's change was to reduce this number of days by one. 
27,759 days divided by 940 gives, for the mean lunar month, 29 
days I2 h 44™ 25|- s , only 22 seconds in excess of the true mean 
length. 

Callippus was probably born about 370 B. c. ; he came to Athens 
about 334 B.C.; the first year of the first of his cycles of j6 years 
was 01. 112, 3 = 330/29 B.C., and probably began on the 29th or 
28th of June. His cycles never apparently came into practical use, 
but they were employed by individual astronomers or chronologists 
for fixing dates ; Ptolemy, for example, gives various dates both 
according to Egyptian reckoning and in terms of Callippic cycles. 2 

Hipparchus's cycle. 

It is only necessary to add that yet another improvement was 
made by Hipparchus about 125 B. C. Ptolemy says of him : 

1 Again, in his work on intercalary months and days, after pre- 
mising that the length of the year is, according to Meton and 
Euctemon, 365J -£ E days, and according to Callippus 365J days 
only, he continues in these words : " We find that the number of 
whole months contained in the 19 years is the same as they make 
it, but that the year in actual fact contains less by -^oth of a day 
than the odd ^ of a day which they give it, so that in 300 years 
there is a deficiency, in comparison with Meton's figure, of 5 days, 
and in comparison with Callippus's figure, of one day." Then, 
summing up his own views in the course of the enumeration of his 
own works, he says : " I have also discussed the length of the year 
in one book, in which I prove that the solar year — that is, the 
length of time in which the sun passes from a solstice to the same 

1 Geminus, Isagoge, 8. 57-60, p. 122. 8-23. 

2 Ptolemy, Syntaxis^ iii. 1, vol. i, p. 196. 6 ; iv. II, vol. i, pp. 344. 14, 345. 
12, 346. 14; v. 3, vol. i, p. 363. 16; vii. 3, vol. ii, pp. 25. 16, 28. 12, 29. 13, 

32.5• 



ch.xix HIPPARCHUS'S CYCLE 297 

solstice again, or from an equinox to the same equinox — contains 
365 days and J, less very nearly -jJo tn °f a day arl d night, and not 
the exact J which the mathematicians suppose it to have in addition 
to the said whole number of days." ' * 

Censorinus gives Hipparchus's period as 304 years, in which there 
are 11 2 intercalary months. 2 Presumably, therefore, Hipparchus 
took four times Callippus's cycle (76 χ 4 = 304) and gave the period 
111,035 days instead of 111,036 ( = 27,759 X 4). This gives, as the 
length of the year, 365 days 5 h $$ m I5-8 3 , while 365J --^Jo days = 
365 days 5 h $5 m 12 s , the excess over the true mean tropic year 
being about 6J minutes. The number of months in the 304 years 
is 304 χ 12 + 28 χ 4 = 3,760, whence the mean lunar month is, accord- 
ing to Hipparchus, 29 days I2 h 44 m 2-§ s , which is very nearly correct, 
being less than a second out in comparison with the present accepted 
figure of 29-53059 days ! 

1 Ptolemy, Syntaxis, iii. 3, vol. i, pp. 207. 7 - 208. 2. 

2 Censorinus, ~De die natali, 18. 9. 



PART II 

ARISTARCHUS OF SAMOS 
ON THE SIZES AND DISTANCES OF THE SUN 

AND MOON 

I 

ARISTARCHUS OF SAMOS 

We are told that Aristarchus of Samos was a pupil of Strato 
of Lampsacus, 1 a natural philosopher of originality, 2 who suc- 
ceeded Theophrastus as head of the Peripatetic school in 288 or 
287 B.C. and held that position for eighteen years. Two other 
facts enable us to fix Aristarchus's date approximately. In 
281/280 B.C. he made an observation of the summer solstice; 3 
and the book in which he formulated his heliocentric hypothesis 
was published before the date of Archimedes' Psammites or Sand- 
reckoner, a work written before 216 B.C. Aristarchus therefore 
probably lived circa 310-230 B.C., that is, he came about 75 years 
later than Heraclides and was older than Archimedes by about 
2$ years. 

Aristarchus was called 4 the mathematician ', doubtless in order 
to distinguish him from the many other persons of the same 
name ; he is included by Vitruvius among the few great men who 
possessed an equally profound knowledge of all branches of science, 
geometry, astronomy, music, &c. ' Men of this type are rare, 
men such as were, in times past, Aristarchus of Samos, Philolaus 
and Archytas of Tarentum, Apollonius of Perga, Eratosthenes 
of Cyrene, Archimedes and Scopinas of Syracuse, who left to 

1 Aetius, i. 15. 5 (Ζλ6\ p. 313). 

2 Galen, His tor. Philos. 3 (D. G. p. 601» 1). 

3 Ptolemy, Syntaxis, iii. 2 (i, pp. 203, 206, ed. Heib.). 



3oo ARISTARCHUS OF SAMOS part π 

posterity many mechanical and gnomonic appliances which they 
invented and explained on mathematical (lit. numerical) and natural 
principles.' 1 That Aristarchus was a very capable geometer is 
proved by his extant work On the sizes and distances of the 
sun and moon, translated in this volume: in the mechanical line 
he is credited with the discovery of an improved sun-dial, the 
so-called σκάφη, which had, not a plane, but a concave hemi- 
spherical surface, with a pointer erected vertically in the middle 
throwing shadows and so enabling the direction and the height 
of the sun to be read off by means of lines marked on the surface 
of the hemisphere. 2 He also wrote on vision, light, and colours. 3 
His views on the latter subjects were no doubt largely influenced 
by his master Strato ; thus Strato held that colours were emanations 
from bodies, material molecules, as it were, which imparted to 
the intervening air the same colour as that possessed by the body, 4 
while Aristarchus said that colours are ' shapes or forms stamping 
the air with impressions like themselves as it were ', 5 that ' colours 
in darkness have no colouring ', 6 and that light is ' the colour 
impinging on a substratum '. 7 It does not appear that Strato 
can be credited with any share in his astronomical discoveries : 
of Strato we are only told (i) that, like Metrodorus before him, 
he held that the stars received their light from the sun (Metrodorus 
alleged this of ' all the fixed stars ', and it is not stated that Strato 
made any limitation) ; 8 (2) that he held a comet to be 4 the light 
of a star enclosed in a thick cloud, just as happens with λαμπτήρες 
(torches) ' ; 9 (3) that, like Parmenides and Heraclitus, he considered 
the heaven to be of fire ; 10 (4) that he regarded time as ' quantity 
in (i.e. expressed by) things in motion and at rest'; 11 (5) that 
he said the divisions of the universe were without limit ; 12 and 
(6) that he maintained that there was no void outside the universe, 
though there might be within it. 13 

1 Vitruvius, De architecture,, i. I. 16. 2 Ibid. ix. 8 (9). I. 

3 Aet. i. 15. 5 {D. G. p. 313), iv. 13. 8 (D. G. pp. 404 and 853). 

4 Aet. iv. 13. 7 (D. G. p. 403). 

5 Ibid. iv. 13. 8 {D. G. pp. 404 and 853). 

6 Ibid. i. 15. 9 (D. G. p. 314). 7 Ibid. i. 15. 5 (Ζλ G. p. 313). 
8 Ibid. ii. 17. 1-2 (D. G. p. 346). 9 Ibid. iii. 2. 4 {D. G. p. 366). 

10 Ibid. ii. 11. 4 (Z>. G. p. 340). n Ibid. i. 22. 4 {D. G. p. 318). 

12 Epiphanius, Adv. haeres. iii. 33 (D. G. p. 5Q2). 

13 Aet. i. 18. 4 .{D. G. p. 316). 



ch. I ARISTARCHUS OF SAMOS 301 

The Heliocentric Hypothesis. 

There is not the slightest doubt that Aristarchus was the first to put 
forward the heliocentric hypothesis. Ancient testimony is unanimous 
on the point, and the first witness is Archimedes, who was a younger 
contemporary of Aristarchus, so that there was no possibility of 
a mistake. Copernicus himself admitted that the theory was 
attributed to Aristarchus, though this does not seem to be generally 
known. Thus Schiaparelli quotes two passages from Copernicus's 
work in which he refers to the opinions of the ancients about the 
motion of the earth. One is in the dedicatory letter to Pope 
Paul III, where Copernicus mentions that he first found out from 
Cicero that one Nicetas (i.e. Hicetas) had attributed motion to the 
earth, and that he afterwards read in Plutarch that certain others 
held that opinion ; he then quotes the Placita, according to 
which ' Philolaus the Pythagorean asserted that the earth moved 
round the fire in an oblique circle, in the same way as the sun 
and moon ', 1 The other passage is in Book I, c. 5, where, after 
an allusion to the views of Heraclides, Ecphantus, and Nicetas 
(Hicetas), who made the earth rotate about its own axis at the 
centre of the universe, he goes on to say that it would not be 
very surprising if any one should attribute to the earth another 
motion besides rotation, namely revolution in an orbit in space ; 
* atque etiam (terram) pluribus motibus vagantem et unam ex astris 
Philolaus Pythagoricus sensisse fertur, Mathematicus non vulgaris.' 
Here, however, there is no question of the earth revolving round the 
sun, and there is no mention of Aristarchus. But it is a curious fact 
that Copernicus did mention the theory of Aristarchus in a passage 
which he afterwards suppressed : i Credibile est hisce similibusque 
causis Philolaum mobilitatem terrae sensisse, quod etiam nonnulli 
Aristarchum Samium ferunt in eadem fuisse sententia.' 2 

I will now quote the whole passage of Archimedes in which 
the allusion to Aristarchus's heliocentric hypothesis occurs, in order 
to show the whole context. 3 

1 Ps. Plutarch, De filac. phil. = Aet. iii. 13. 2 (D. G. p. 378). 

2 De Revolutionibus Caelestibus, ed. Thorun., 1873, p. 34 note, quoted in 
Gomperz, Griechische Denker, i 3 , p. 432. 

3 Archimedes, ed. Heiberg, vol. ii, p. 244 (Arenarius I. 4-7); The Works of 
Archimedes, ed. Heath, pp. 221, 222. 



3θ2 ARISTARCHUS OF SAMOS part π 

'You are aware ['you' being King- Gelon] that "universe" is 
the name given by most astronomers to the sphere, the centre 
of which is the centre of the earth, while its radius is equal to 
the straight line between the centre of the sun and the centre of 
the earth. This is the common account (τα γραφόμβνα), as you 
have heard from astronomers. But Aristarchus brought out a book 
consisting of certain hypotheses, wherein it appears, as a con- 
sequence of the assumptions made, that the universe is many 
times greater than the " universe " just mentioned. His hypotheses 
are that the fixed stars and the sun remain unmoved, that the 
earth revolves about the sun in the circumference of a circle, the 
sun lying in the middle of the orbit, and that the sphere of the fixed 
stars, situated about the same centre as the sun, is so great that 
the circle in which he supposes the earth to revolve bears such 
a proportion to the distance of the fixed stars as the centre of 
the sphere bears to its surface. Now it is easy to see that this is 
impossible ; for, since the centre of the sphere has no magnitude, 
we cannot conceive it to bear any ratio whatever to the surface 
of the sphere. We must, however, take Aristarchus to mean this : 
since we conceive the earth to be, as it were, the centre of the 
universe, the ratio which the earth bears to what we describe as 
the " universe " is equal to the ratio which the sphere containing 
the circle in which he supposes the earth to revolve bears to the 
sphere of the fixed stars. For he adapts the proofs of the pheno- 
mena to a hypothesis of this kind, and in particular he appears 
to suppose the size of the sphere in which he makes the earth 
move to be equal to what we call the " universe ".' 

We shall come back to the latter part of this passage ; at present 
we are concerned only with the italicized words. The heliocentric 
hypothesis is stated in language which leaves no room for dispute 
as to its meaning. The sun, like the fixed stars, remains unmoved 
and forms the centre of a circular orbit in which the earth revolves 
round it; 1 the sphere of the fixed stars has its centre at the 

1 There is only one slight awkwardness in the language. The sentence is 

υποτίθεται yap τα μεν άπλανεα των (ίστρων κα\ τον αλιον μενειν άκίνητον, τάν δε γάν 
περι,φέρεσθαι περί τον αλιον κατά κύκλου περιφέρειαν, 6s εστίν εν μέσω τω δρόμω 
κείμενος, and it would be natural to suppose that the relative δς would refer to 
the masculine substantive nearest to it, i.e. κύκλου, 'circle,' rather than τον άλιον, 
'the sun 5 ; but ' which is situated in the middle of the (earth's) course'' cannot 
possibly refer to the circle, i. e. to the course itself, and must refer to the sun. 
The awkwardness suggested to Bergk {Fiinf Abhandlungen, 1883, p. 162) that 
Archimedes wrote δς εστίν εν μέσω τω ούρανω, 'which is in the middle of the 
heaven} This would enable δς to refer to the ' circle ', but there seems to be no 
sufficient ground for changing the reading δρόμω. 



ch.i ARISTARCHUS OF SAMOS 303 

centre of the sun. But a question arises as to the form in which 
Aristarchus's hypotheses were given out. The expression used 
by Archimedes is ύποθεσίων τίνων εξέδωκβν γραφάς, 'put out 
γραφαί of certain hypotheses.' I take it in the sense of bringing• 
out a tract or tracts consisting of or stating certain hypotheses ; 
for one of the meanings of the word γραφή is a ( writing ' or 
a written * description '. Heiberg takes γραφαί in this sense, but 
regards ύποθεσίων as the title of the book (' libros quosdam edidit, 
qui hypotheses inscribuntur ' 1 ). Hultsch, 2 however, takes γραφαί 
in its other possible sense of * drawings ' or figures constructed 
to represent the hypotheses ; and Schiaparelli 3 suggests that the 
word γραφή here used seems not only to signify a verbal description 
but to include also the idea of explanatory drawings. I agree 
that it is probable enough that Aristarchus's tract or tracts included 
geometrical figures illustrating the hypotheses, but I still think 
that the word γραφαί here does not itself mean • figures' but 
means written statements of certain hypotheses. This seems to 
me clear from the words immediately following γραφάς, namely 
kv αϊ? e/c των υποκειμένων συμβαίνει κ.τ.ί., 4 in which it results 
from the assumptions made that the universe is many times greater 
than our " universe " above mentioned ' ; 4 in which ' can only refer 
to γραφά? or ύποθβσίων, and it cannot refer to ύποθςσίων because 
what follows from the assumptions made cannot be in those 
assumptions which are nothing but the hypotheses themselves; 
therefore ' in which ' refers to γραφάς, but a result following from 
assumptions does not follow in figures illustrating those assump- 
tions but in the course of a description of them or an argument 
about them. The words ' in which it results . . .' also show clearly 
enough that the tract or tracts did not merely state the hypothesis 
but also included some kind of geometrical proof. 4 I need only 

1 Archimedes, ed. Heiberg, vol. ii, p. 245. 

2 Hultsch, art. 'Aristarchus 5 in Pauly-Wissowa's Real-Encyclopadie^'u^. 875. 

3 Schiaparelli, Origine del sistema filanetario eliocentrico ftresso i Greet, p. 95. 

4 Bergk (Fiinf Abhandlungen, p. 160) thinks that 'γραφαί cannot be taken as 
synonymous with γράμματα : this would be a somewhat otiose circumlocution : 
but it means the "outline" (Umriss), like καταγραφή. Archimedes chooses this 
expression because Aristarchus had rather indicated his hypotheses than worked 
them out and established them.' I do not think this inference necessary ; 
γραφάς may be quite colourless without being otiose, a sufficient reason for its 
insertion being the fact that some word other than νποθεσίων is necessary as an 



304 ARISTARCHUS OF SAMOS partii 

add that there are other cases of the use of γραφή in the sense 
of ' writing- ' ; cf. an expression in Eutocius, ' I have come across 
writings (γραφαΐς) of many famous men which give this problem ' 
[that of the two mean proportionals]. 1 

Our next evidence is a passage of Plutarch 2 : 

* Only do not, my good fellow, enter an action against me for 
impiety in the style of Cleanthes, who thought it was the duty 
of Greeks to indict Aristarchus of Samos on the charge of impiety 
for putting in motion the Hearth of the Universe, this being the 
effect of his attempt to save the phenomena by supposing the 
heaven to remain at rest and the earth to revolve in an oblique 
circle, while it rotates, at the same time, about its own axis.' 

Here we have the additional detail that Aristarchus followed 
Heraclides in attributing to the earth the daily rotation about its 
axis; Archimedes does not state this in so many words, but it 
is clearly involved by his remark that Aristarchus supposed that 
the fixed stars as well as the sun remain unmoved in space. When 
Plutarch makes Cleanthes say that Aristarchus ought to be indicted 
for the impiety of l putting the Hearth of the Universe in motion ', 
he is probably quoting the exact words used by Cleanthes, who 
doubtless had in mind the passage in Plato's Phaedrus where 

* Hestia abides alone in the House of the Gods '. A similar ex- 
pression is quoted by Theon of Smyrna from Dercyllides, who 

• says that we must suppose the earth, the Hearth of the House 
of the Gods according to Plato, to remain fixed, and the planets 
with the whole embracing heaven to move, and rejects with 
abhorrence the view of those who have brought to rest the things 
which move and set in motion the things which by their nature and 
position are unmoved, such a supposition being contrary to the 
hypotheses of mathematics ' ; 3 the allusion here is equally to 
Aristarchus, though his name is not mentioned. A tract ' Against 
Aristarchus ' is mentioned by Diogenes Laertius among Cleanthes' 
works ; and it was evidently published during Aristarchus's lifetime 
(Cleanthes died about 232 B.C.). 

antecedent to the relative sentence ' in which it follows from the assumptions 
made, &c.' 

1 Archimedes, ed. Heiberg, vol. iii, p. 66. 9. 

2 Plutarch, De facie in orbe lunae, c. 6, pp. 922 F - 923 A. 

3 Theon of Smyrna, p. 200. 7-12, ed. Hiller. 



ch.i ARISTARCHUS OF SAMOS 305 

Other passages bearing on our present subject are the fol- 
lowing. 

4 Aristarchus sets the sun among the fixed stars and holds that 
the earth moves round the sun's circle (i.e. the ecliptic) and is put in 
shadow according to its (i. e. the earth's) inclinations.' x 

One of the two versions of this passage has * the disc is put 
in shadow ', and it would appear, as Schiaparelli says, 4 that the 
words u the disc " were interpolated by some person who thought 
that the passage was an explanation of solar eclipses.' It is indeed 
placed under the heading 4 Concerning the eclipse of the sun ' ; 
but this is evidently wrong, for we clearly have here in the 
concisest form an explanation of the phenomena of the seasons 
according to the system of Copernicus. 2 

4 Yet those who did away with the motion of the universe and 
were of opinion that it is the earth which moves, as Aristarchus 
the mathematician held, are not on that account debarred from 
having a conception of time.' 3 

4 Did Plato put the earth in motion, as he did the sun, the moon, 
and the five planets, which he called the instruments of time on 
account of their turnings, and was it necessary to conceive that the 
earth u which is globed about the axis stretched from pole to pole 
through the whole universe " was not represented as being held 
together and at rest, but as turning and revolving (στρεφομζνην 
καϊ άνειλουμένην), as Aristarchus and Seleucus afterwards main- 
tained that it did, the former stating this as only a hypothesis 
(υποτιθέμενο? μόνον), the latter as a definite opinion (και απο- 
φαινόμενος) ? ' 4 

4 Seleucus the mathematician, who had written in opposition to 
the views of Crates, and who himself too affirmed the earth's motion, 
says that the revolution {περιστροφή) of the moon resists the rota- 
tion [and the motion] 5 of the earth, and, the air between the two 
bodies being diverted and falling upon the Atlantic ocean, the sea 
is correspondingly agitated into waves.' 6 

When Plutarch refers to Aristarchus as only putting forward the 
double motion of the earth as a hypothesis, he must presumably 

1 Aet. ii. 24. 8 (D. G. p. 355. 1-5). 

2 Schiaparelli, I firecui-sori, p. 50. 

3 Sextus Empiricus, Adv. Mathematicos, x. 174, p. 512. 19, Bekker. 

4 Plutarch, Plat, quaest. viii. 1, 1006 C. 

δ The Ps. Plutarch version has the words και τη κίνησα, * and the motion,' 
after αυτψ τη δίν>ι φησί; Stobaeus omits them, and has τω δίνω instead of rij δύ>?/. 
6 Aet. iii. 17. 9 (D. G. p. 383). 

1410 X 



3o6 ARISTARCHUS OF SAMOS part π 

be basing himself on nothing more than the word hypotheses used 
by Archimedes, and his remark does not therefore exclude the 
possibility of Aristarchus having supported his hypothesis by some 
kind of argument ; nor can we infer from Plutarch that Seleucus 
went much further towards proving it. Plutarch says that 
Seleucus declared the hypothesis to be true (αποφαινομένου) , but 
it is not clear how he could have attempted to prove it. Schiapa- 
relli suggests that Aristarchus 's attitude may perhaps be explained 
on the basis of the difference between the roles of the astronomer 
and the physicist as distinguished by Geminus in the passage quoted 
above (pp. <ιη$-6). Aristarchus, as the astronomer and mathe- 
matician, would only be concerned to put forward geometrical 
hypotheses capable of accounting for the phenomena; he may 
have left it to the physicists to say 'which bodies ought from 
their nature to be at rest and which to move \ But this is only 
a conjecture. 

Seleucus, of Seleucia on the Tigris, is described by Strabo 1 as 
a Chaldaean or Babylonian ; he lived about a century after Aris- 
tarchus and may have written about 150 B. c. The last of the 
above quotations is Aetius's summary of his explanation of 
the tides, a subject to which Seleucus had evidently given much 
attention ; 2 in particular, he controverted the views held on this 
subject by Crates of Mallos, the i grammarian ', who wrote on 
geography and other things, as well as on Homer. The other 
explanations of the tides summarized by Aetius include those ot 
Aristotle and Heraclides, who sought the explanation in the sun, 
holding that the sun sets up winds, and that these winds, when 
they blow, cause the high tide and, when they cease, the low tide ; 
Dicaearchus who put the tides down to the direct action of the 
sun according to its position ; Pytheas and Posidonius who con- 
nected them with the moon, the former directly, the latter through 
the setting up of winds ; Plato who posited a certain general oscil- 
lation of the waters, which pass through a hole in the earth ; 3 
Timaeus who gave as the reason the unequal flow of rivers from 
the Celtic mountains into the Atlantic ; then, immediately before 
Seleucus, are mentioned Crates ' the grammarian ' and Apollo- 

1 Strabo, xvi. 1. 6, i. 1. 9. 2 Cf. Strabo, iii. 5. 9. 

3 Cf. Phaedo 1 1 1 C sqq. 



ch.i ARISTARCHUS OF SAMOS 307 

dor us of Corcyra, the account of whose views is vague enough, 
the former attributing the tides to 'the counter- movement (αντι- 
σπασμός) of the sea,' 1 and the latter to ' the refluxes from the Ocean '. 
When Aetius adds, in introducing Seleucus's views, that • he too 
made the earth move ', we should expect that he had just before 
mentioned some one else who had done the same. But Crates 
adhered to the old view and did not make the earth move ; 2 nor is 
there anything to suggest that Apollodorus attributed motion to 
the earth. Consequently Bergk supposes that, just before the 
notice of Seleucus's explanation of the tides with reference to 
the earth's motion, there must have been a notice of a different 
explanation of them by a person who also attributed motion to the 
earth, and that, as we know of no other person by name who 
adopted Aristarchus's views, except Seleucus, the notice which has 
dropped out must have given a different explanation of the tides 
by Aristarchus himself. 3 But, as the motion of the earth referred 
to in Seleucus's explanation may be rotation only {δίνη or δΐνος), it 
seems possible that Heraclides (who made the earth rotate) is the 
other person referred to in the collection of notices as having 
4 made the earth move ', although he is mentioned some way back. 
To judge by Seleucus's explanation of the tides, he would seem to 
have supposed that the atmosphere about the earth extended as 
far as the moon and rotated with the earth in 24 hours, and that 
the resistance of the moon acted upon the rotating atmosphere 
either by virtue of the relative slowness of the moon's revolution 
about the earth or of its motion perpendicular to the equator ; 4 
Strabo tells us that Seleucus had discovered periodical inequalities 
in the flux and reflux of the Red Sea which he connected with the 
position of the moon in the zodiac. 5 

No one after Seleucus is mentioned by name as having accepted 
the doctrine of Aristarchus, and if other Greek astronomers refer to 
it, they do so only to denounce it, as witness Dercyllides. 6 The 
rotation of the earth is, however, mentioned as a possibility by 
Seneca. 

1 Some details of Crates* views are also given in Strabo, i. 1.8. 

2 Bergk (Fiinf Abhandlungen, p. 166) quotes from Strabo, i. 2. 24, the words 
την πάρυδον τον ηλίου. 3 Bergk, op. cit., p. 167. 

4 Schiaparelli, I pre curs or i, p. 36. 5 Strabo, iii. 5. 9. 

6 Theon of Smyrna, p. 200. 7-12 : see above (p. 304). 

X % 



3o8 ARISTARCHUS OF SAMOS part π 

* It will be proper to discuss this, in order that we may know 
whether the universe revolves and the earth stands still, or the 
universe stands still and the earth rotates. For there have been 
those who asserted that it is we whom the order of nature causes to 
move without our being aware of it, and that risings and settings do 
not occur by virtue of the motion of the heaven, but that we ourselves 
rise and set. The subject is worthy of consideration, in order that 
we may know in what conditions we live, whether the abode allotted 
to us is the most slowly or the most quickly moving, whether God 
moves everything around us, or ourselves instead.' x 

Hipparchus, himself a contemporary of Seleucus, reverted to the 
geocentric system, and it was doubtless his great authority which 
sealed the fate of the heliocentric hypothesis for so many centuries. 
^The reasons which weighed with Hipparchus were presumably (in 
addition to the general prejudice in favour of maintaining the earth 
in the centre of the universe) the facts that the system in which the 
earth revolved in a circle of which the sun was the exact cenire 
failed to ' save the phenomena ', and in particular to account for the 
variations of distance and the irregularities of the motions, which 
became more and more patent as methods of observation improved ; 
that, on the other hand, the theory of epicycles did suffice to repre- 
sent the phenomena with considerable accuracy ; and that the latter 
theory could be reconciled with the immobility of the earth. 

We revert now to the latter part of the passage quoted above 
from Archimedes, in which he comments upon the assumption of 
Aristarchus that the sphere of the fixed stars is so great that the 
ratio in which the earth's orbit stands to the said sphere is such 
a ratio as that which the centre of the sphere bears to its surface. 
If this is taken in a strictly mathematical sense, it means of course 
that the sphere of the fixed stars is infinite in size, a supposition 
which would not suit Archimedes' purpose, because he is under- 
taking to prove that he can evolve a system for expressing large 
numbers which will enable him to state easily in plain words the 
number of grains of sand which the whole universe could contain ; 
hence, while he wishes to base his estimate of the maximum size of 
the universe upon some authoritative statement which will be 
generally accepted, and takes the statement of Aristarchus as suit- 

1 Seneca, Nat. Quaes/, vii. 2. 3. * 



ch.i. ARISTARCHUS OF SAMOS 309 

able for his purpose, he is obliged to interpret it in an arbitrary- 
way which he can only justify by somewhat sophistically pressing 
the mathematical point that Aristarchus could not have meant to 
assert that the sphere of the fixed stars is actually infinite in size 
and therefore could not have wished his statement to be taken 
quite literally ; consequently he suggests that a reasonable inter- 
pretation would be to take it as meaning that 

(diameter of earth) : (diameter of ' universe ') = 

(diam. of earth's orbit) : (diam. of sphere of fixed stars), 

instead of 

: (surface of sphere of fixed stars) = 

(diam. of earth's orbit): (diam. of sphere of fixed stars), 

and he explains that the ' universe ' as commonly conceived by the 
astronomers of his time (he refers no doubt to the adherents of 
the system of concentric spheres) is a sphere with the earth as 
centre and radius equal to the distance of the sun from the earth, 
and that Aristarchus seems to regard the sphere containing (as a 
great circle) the orbit in which the earth revolves about the sun as 
equal to the ' universe ' as commonly conceived, so that the second 
and third terms of the first of the above proportions are equal. 

While it is clear that Archimedes' interpretation is not justified, 
it may be admitted that Aristarchus did not mean his statement to 
be taken as a mathematical fact. He clearly meant to assert no 
more than that the sphere of the fixed stars is incomparably greater 
than that containing the earth's orbit as a great circle ; and he was 
shrewd enough to see that this is necessary in order to reconcile 
the apparent immobility of the fixed stars with the motion of 
the earth. The actual expression used is similar to what was 
evidently a common form of words among astronomers to ex- 
press the negligibility of the size of the earth in comparison with 
larger spheres. Thus, in his own tract On the sizes and distances 
of the sun and moon, Aristarchus lays down as one of his assump- 
tions that ' the earth is in the relation (λόγοι/ εχειν) of a point and 
centre to the sphere in which the moon moves '. In like manner 
Euclid proves, in the first theorem of his Phaenomena, that ' the 
earth is in the middle of the universe (κόσμος) and holds the 



3io ARISTARCHUS OF SAMOS part π 

position (τάξιν) of centre relatively to the universe '. Similarly 
Geminus l describes the earth as ' in the relation of a centre to the 
sphere of the fixed stars'; Ptolemy 2 says that the earth is not 
sensibly different from a point in relation to the radius of the 
sphere of the fixed stars ; according to Cleomedes 3 the earth is 
1 in the relation of a centre ' to the sphere in which the sun moves, 
and a fortiori to the sphere of the fixed stars, but not to the 
sphere in which the moon moves. 

In Aristarchus's extant treatise On the sizes and distances of the 
sun and moon there is no hint of the heliocentric hypothesis, but 
the sun and moon are supposed to move in circles round the earth 
as centre. From this we must infer either (i) that the work in 
question was earlier than the date at which he put forward the 
hypotheses described by Archimedes, or (2) that, as in the tract the 
distances of the sun from the earth and of the moon from the earth 
are alone in question, and therefore it was for the immediate pur- 
pose immaterial which hypothesis was taken, Aristarchus thought 
it better to proceed on the geocentric hypothesis which was familiar 
to everybody. Schiaparelli 4 suggests that one of the reasons which 
led Aristarchus to place the sun in the centre of the universe was 
probably the consideration of the sun's great size in comparison 
with the earth. Now in the treatise referred to Aristarchus finds 
the ratio of the diameter of the sun to the diameter of the earth to 
lie between 19:3 and 43 : 6 ; this makes the volume of the sun 
something like 300 times the volume of the earth, and, although 
the principles of dynamics were then unknown, it might even in 
that day seem absurd to make the body which was so much larger 
revolve round the smaller. 

There is no reason to doubt that, in his heliocentric system, 
Aristarchus retained the moon as a satellite of the earth revolving 
round it as a centre; thus even in his system there was one epi- 
cycle, that described by the moon about the earth as centre. 

1 Geminus, Isagoge, 18. 16, p. 186. 16, ed. Manitius. 

2 Ptolemy, Syntaxis, i. 6, p. 20. 5, Heib. 

3 Cleomedes, De motu circularly i. 11. 4 Schiaparelli, I precursori, p. 33. 



ch.i ARISTARCHUS OF SAMOS 311 

The apparent diameter of the sun. 

Another passage of the Sand-reckoner of Archimedes states that 

1 Aristarchus discovered that the sun's apparent size is about one 
720th part of the zodiac circle.' λ 

This, again, is a valuable contribution to our knowledge of 
Aristarchus, for in the treatise On the sizes and distances of the 
sun and moon he makes the apparent diameter not T 4o tn of the 
zodiac circle, or -|°, but one-fifteenth part of a sign, that is to say 2 , 
which is a gross over-estimate. The nearest estimate to this which 
we find recorded appears to be that mentioned by Macrobius, 2 who 
describes an experiment made with a hemispherical dial by marking 
the points on which the shadow of the upright needle fell at the 
moments respectively when the first ray of the sun as it began to 
rise fell on the instrument and when the sun just cleared the horizon 
respectively. The result showed that the interval of time was |th 
of an hour, which gave as the apparent diameter of the sun 2T e tn 
of 360 or i§°. Macrobius would apparently have us believe that 
this very inaccurate estimate was due to the Egyptians. We have, 
however, seen reason to believe that Macrobius probably attributed 
to the ' Egyptians ' the doctrines of certain Alexandrian astro- 
nomers, 3 and in the present case it would seem that we have to do 
with an observation very unskilfully made by some even less com- 
petent person. 4 The Babylonians had, however, many centuries 
before arrived at a much closer approximation ; they made the 
time which the sun takes to rise ^ο ίη °f an hour, and, even if 
the hour is the double hour (one- twelfth of a day and night), this 
gives i° as the apparent diameter of the sun. How, then, did 
Aristarchus in his extant work come to take 2° as the value ? 
Tannery has an interesting suggestion, which is however perhaps 
too ingenious. 5 ' If Aristarchus chose for the apparent diameter 
of the sun a value which he knew to be false, it is clear that his 
treatise was mainly intended to give a specimen of calculations 

1 Archimedes, ed. Heiberg, ii, p. 248. 19; The Works of Archimedes, ed. 
Heath, p. 223. 

2 Macrobius, In somn. Scift. i. 20. 26-30. 3 See p. 259 above. 

4 Hultsch, Poseidonios iiber die Grbsse und Entfernung der Sonne, p. 43. 

5 Tannery, 'Aristarque de Samos' in Mem. de la Soc. des sciences phys. et 
nat. de Bordeaux, 2 e ser. v. 1883, p. 241 ; Memoires scientifiques, ed. Heiberg 
and Zeuthen, i, pp. 375-6. 



3i2 ARISTARCHUS OF SAMOS partii 

which require to be made on the basis of more exact experimental 
observations, and to show at the same time that, for the solution 
of the problem, one of the data could be chosen almost arbitrarily. 
He secured himself in this way against certain objections which 
might have been raised. According to the testimony of Macrobius, 
it seems that in fact the Egyptians had, by observations completely 
erroneous, fixed the apparent diameter of the sun at ^y^h °^ tne 
circumference, i. e. if °. Aristarchus seems to have deliberately 
chosen to assign it a still higher value ; but it is beyond question 
that he was perfectly aware of the consequences of his hypothesis.' 
Manitius 1 suggests that the One-fifteenth part (πβι/τβκαίδίκατον 
μίροξ) ' of a sign of the zodiac in Aristarchus's treatise should be 
altered into ' one-fiftieth part ' (π^ντηκοστον /jepoy), which would 
give the quite acceptable value of o° 36'. But the propositions in 
the treatise in which the hypothesis is actually used seem to make 
it clear that ' one-fifteenth' is what Aristarchus really wrote. Unless 
therefore we accept Tannery's suggestion, we seem to be thrown 
back once more on the supposition that the treatise was an early 
work written before Aristarchus had made the more accurate 
observation recorded by Archimedes. From the statement of 
Archimedes that Aristarchus discovered (ςνρηκότος) the value of 
T |yth, I think we may infer with safety that Aristarchus was at 
least the first Greek who had given it, and we have therefore an 
additional reason for questioning the tradition which credits Thales 
with the discovery. How Aristarchus obtained his result we are 
not told, but, seeing that he is credited with the invention of an 
improved sun-dial (σκάφη), it is possible that it was by means of 
this instrument that he made his observations. Archimedes himself 
seems to have been the first to think of the better method of using 
an instrument for measuring angles ; by the use of a rough instru- 
ment of this kind he made the apparent angular diameter of the sun 
lie between the limits of y^th and atroth of a right angle. Hippar- 
chus used for the same purpose a more elaborate instrument, his 
dioptra, the construction of which is indicated by Ptolemy, 2 and 
is more fully described by Pappus in his commentary on Book V of 

1 Proclus, Hypotyposis, ed. Manitius, note on p. 292. 

2 Ptolemy, Syntaxis, v. 14, p. 417. 2-3 and 20-23, e< ^• Heib. 



ch.i ARISTARCHUS OF SAMOS 313 

Ptolemy, quoted by Theon of Alexandria ; l Proclus describes it 
somewhat differently. 2 Though we gather that Hipparchus made 
many observations of the apparent diameters of the sun and 
moon, 3 only one actual result is handed down ; he found that the 
diameter of the moon was contained about 650 times in the circle 
described by it. 4 This would no doubt be the mean of the different 
observations of the moon at its varying distances ; it is of course 
equivalent to nearly o° 33' 14". Ptolemy complains that the 
requisite accuracy could not be secured by the dioptra ; he there- 
fore checked the observations as regards the moon by means of 
1 certain lunar eclipses ', and found Hipparchus 's values appreciably 
too high. Ptolemy 5 himself made the apparent diameter of the 
moon to be (a) at the time when it is furthest from the earth 
o° 31' 20", and (δ) at its least distance o° 35' 20". The mean of 
these figures being o° 33' 20", and the true values corresponding 
to Ptolemy's figures being 29/ 26" and 32/ 51", it follows that 
Hipparchus's mean value is actually nearer the true mean value than 
Ptolemy's. 6 Aristarchus, as we shall see, took the apparent dia- 
meters of the sun and moon to be the same. Sosigenes (2nd c. A. D.) 
showed that they are not always equal by adverting to the pheno- 
menon of annular eclipses of the sun, 7 and doubtless Hipparchus 
had observed the differences ; Ptolemy found that the apparent 
diameter of the sun was approximately constant, whenever observed, 
its value being the same as that of the moon when at its greatest 
distance, not (' as supposed by earlier astronomers ') when at its 
mean distance. 8 Another estimate of the apparent diameter of the 
sun, namely T -Joth °f tne complete circle described by the sun, or 
29/, is given by Cleomedes as having been obtained by means of 
a water-clock ; he adds that the Egyptians are said to have been the 
first to discover this method. 9 Yet another valuation appears in 

1 Theon, in Ptolem. ??iagn. construct, p. 262. 

2 Proclus, Hypotyposis, ed. Manitius, pp. 126. 13- 128. 13. 

3 Ptolemy, Syntaxis, loc. cit. 

4 Ptolemy, Syntaxis, iv. 9, p. 327. 1-3, Heib. 

5 Ptolemy, v. 14, p. 421. 3-5 ; Pappus, ed. Hultsch, vi, p. 556. 17-19. 

6 On the whole of this subject see Hultsch, ' Winkelmessungen durch die 
Hipparchische Dioptra' in Abhandlungen zur Gesch. d. Math, ix (Cantor- 
Festschrift), 1899, pp. 193-209. 

7 Simplicius on De caeto, p. 505. 7-9, Heib. 

8 Ptolemy, Syntaxis, v. 14, p. 417. 3-11, Heib. 

9 Cleomedes, De motu circulari, ii. 1, pp. 136-8, ed. Ziegler. 



3i4 ARISTARCHUS OF SAMOS part π 

Martianus Capella ; * the diameter of the moon is there estimated as 
-^o tft of i ts orbit or 36'. This estimate was probably quoted from 
Varro, and belongs to a period anterior to Hipparchus. 2 

The Year and the Great Year of Aristarchus. 

We are told by Censorinus that Aristarchus added x^^rd of a 
day to Callippus's figure of 365J days for the solar year, 3 and that 
he gave 2,484 years as the length of the Great Year, or the period 
after which the sun, the moon, and the five planets return to the 
same position in the heavens. 4 Tannery 5 shows that 2,484 years 
is probably a mistake for 2,434 years, and he gives an explanation, 
which seems convincing, of the way in which Aristarchus arrived 
at his figures. They were doubtless derived from the Chaldaean 
period of 223 lunations and the multiple of this by 3, which was 
called β^λιγμός, a period defined by Geminus as the shortest time 
containing a whole number of days, a whole number of months 
(synodic), and a whole number of anomalistic months. 6 The Greeks 
were by Aristarchus 's time fully acquainted with these periods, 
which were doubtless known through the Chaldaean Berosus, 
who flourished about 280 B.C., in the time of Alexander the Great, 
and founded an astronomical school on the island of Cos opposite 
Miletus. Ptolemy, 7 too, says of the first of the two periods (which 
he attributes to ' the ancients ', not the Chaldaeans specifically) that 
it was estimated at 6,5 8 5§ days, containing 223 lunations, 239 
* restorations of anomaly ' (i. e. anomalistic months), 242 4 restora- 
tions of latitude ' (i.e. draconitic months, the draconitic month — a 
term not used by Ptolemy — meaning the period after which the 
moon returns to the same position with respect to the nodes), and 
241 sidereal revolutions plus io§° which the sun describes in the time 
in addition to 18 sidereal revolutions. The exeligmus then, which 
was three times this period, consisted of 19,756 days, containing 
669 lunations, 717 anomalistic months, 726 draconitic months, and 

1 Martianus Capella, De nuptiis fthilologiae et Mercurii, viii. 860. 

2 Tannery, Recherches sur Vhistoire de Pastronomie ancie?me ) p. 334. 

3 Censorinus, De die natali, c. 19. 2. 4 Ibid., c. 18. II. 

5 Tannery, ' La Grande Annee d'Aristarque de Samos ' in Mem. de la Soc. 
des sciences phys. et naturelles de Bordeaux, 3 e serie, iv. 1888, pp. 79-96. 

6 Geminus, Isagoge, c. 18, pp. 200 sqq.^ed. Manitius. 

7 Ptolemy, Syntaxis, iv. 2, pp.269. 21-270. 18, Heib. 



CH.i ARISTARCHUS OF SAMOS 315 

723 sidereal revolutions plus 32 described by the sun in the period 
over and above 54 sidereal revolutions. 

It follows that the number of days in the sidereal year is 

Ϊ975* 5 _ J 9756 _ 45•*975 6 „ 889020 _ _ 3 _ 

54 + A 2 o " 54 + A " 2434 *434 " 3 54 + 4 ° 68 ' 

Now - 4 - 8 g 6 -£ = 1623 — •|. Thus, in replacing• the complementary 
4H 3 68 by tews Aristarchus followed the fashion of only admitting 
fractions with unity as numerator, and thereby only neglected 
the insignificant fraction 1623 1 4868 or 790 1 0764 • 

It is clear that Aristarchus multiplied by 45 so as to avoid all 
fractions, and so arrived at 889,020 days containing 2,434 sidereal 
years, 30,105 lunations, 32,265 anomalistic months, 32,670 draconitic 
months, and 32,539 sidereal months. 

Tannery gives good reason for thinking that this evaluation of 
the solar year at 365J γ^-^ days was really a sort of argument in 
a circle and was therefore worthless. The Chaldaean period was 
obtained from the observation of eclipses ; those which were similar 
were classified, and it was recognized that they returned at the end 
of a period estimated at 6,585^ days. If the theory of the sun had 
been sufficiently established, or if the difference of longitude between 
the positions of two similar eclipses had been observed and allow- 
ance made for the solar anomaly, it would have been possible to 
evaluate with precision the number of degrees traversed during 
the period by the sun over and above the whole number of its 
revolutions. But this precision was beyond the powers of the 
Chaldaeans, and the estimate of the excess of io|° was probably 
obtained by means of the simple difference between 6585^ days and 
18 years of 365 \ days or 6,574^ days. This difference is iof days, 
and, if this is turned into degrees by multiplying by 360/365J, we 
have about iof ^ τ ; the complementary fraction ^ would then be 
neglected as unimportant. Thus Aristarchus's estimate of 365 J i-^s 
days was valueless, as the Chaldaean period itself depended on 
a solar year of 365^ days. 

The question remaining is whether Aristarchus's Great Year was 
intended to be the period which brings all the five planets as well 
as the sun and moon back again to the same places, as appears to 
be implied by Censorinus, who mentions different estimates of the 



3i6 ARISTARCHUS OF SAMOS 

Great Year (including Aristarchus's) just after an explanation that 
' there is also a year which Aristotle calls the greatest rather than 
the great year, which is completed by the sun, the moon, and the five 
planets when they return together to the same sign in which they 
were once before simultaneously found '. As Tannery observes, if 
Aristarchus's Great Year corresponded to an effective determination 
of the periods of the revolutions of the planets, it would have 
a particular interest because Aristarchus would have, in accordance 
with his system, to treat the revolution of Mercury and Venus as 
heliocentric, whereas in the earlier estimates of Great Years, e.g. 
that of Oenopides, the revolution of these planets was geocentric 
and of the same mean duration as that of the sun, so that they could 
be left out of account. But, just as we were obliged to conclude 
that Oenopides could not have maintained that his Great Year of 
59 years contained a whole number of the periods of revolution of 
the several planets, so it seems clear that Aristarchus could hardly 
have maintained that his Great Year exactly covered an integral 
number of the periods of revolution of the nvQ planets. For suppose 
that his Great Year of 889,020 days is divided by the respective 
periods of their sidereal revolutions, and that we take the nearest 
whole numbers to the quotients — say 10,106 for Mercury, 3,950 for 
Venus, 1 ,294 for Mars, 206 for Jupiter, 83 for Saturn — the errors 
as regards the positions at the end of the period would amount, 
according to Tannery's calculation, to 133 for Saturn, 70 for 
Jupiter, 25 for Mars, 171 for Venus, and n° for Mercury. This 
being so, it is difficult to believe that the period of Aristarchus is 
anything more than a luni-solar cycle. 1 

1 Tannery, loc. cit., pp. 93-4. 



II 



ARISTARCHUS ON THE SIZES AND DISTANCES OF 

THE SUN AND MOON 

History of the Text and Editions. 

At the beginning of Book VI of his Synagoge, Pappus refers to 
want of judgement (as to what to include and what to omit) on the 
part of ' many of those who teach the Treasury of Astronomy (τον 
άστρονομούμβνον τόπον)' . The marginal note of the contents of the 
Book, written in the third hand in the oldest MS., says that it 
contains solutions of difficulties kv τω μι κ ρω άστρονομονμζνω, which 
words, with τόπω understood, indicate that the collection of trea- 
tises referred to by Pappus was known as the ' Little Astronomy ', 
as we might say. The collection formed a sort of preliminary 
course, introductory to what would presumably be regarded as the 
* Great Astronomy ', the Syntaxis of Ptolemy. From Pappus's own 
references in the course of Book VI we may infer that the Little 
Astronomy certainly included the following books : 

Autolycus, On the moving sphere (περί κινούμενης σφαίρας). 
Euclid, Optics, 

„ Phaenomena. 

Theodosius, Sphaerica, 

„ On days and nights. 

Aristarchus, On the sizes and distances of the sun and moon. 

No doubt Autolycus's other treatise, On risings and settings, 
Theodosius's On habitations, and Hypsicles' αναφορικός (De ascen- 
sionibus) were also included ; they duly appear in MSS. containing 
the whole collection. 1 All these treatises are extant in Greek as 
well as in Arabic. Not so another important work, the Sphaerica 

1 Heiberg, Literargeschichtliche Studien iiber Euklid, 1882, p. 152. 



3i8 TREATISE ON SIZES AND DISTANCES partii 

of Menelaus, which has only survived in the Arabic and in transla- 
tions therefrom, but seems to have belonged to the collection, since 
Pappus gives four propositions found in Menelaus; 1 this treatise 
was important for the study of the Syntaxis, as is proved by the 
fact that Ptolemy takes for granted certain propositions of 
Menelaus. 2 

As regards some of these treatises it is certain that they were by 
no means the first or the only works dealing with the same subjects. 
Thus Euclid's Phaenomena is closely akin to Autolycus's On the 
moving sphere, and both assume as well known a number of 
propositions which are found in Theodosius's Sphaerica, a work 
much later in date. 3 It is certain therefore that before the date of 
Autolycus (latter half of fourth century B. C.) there was in existence 
a body of sphaeric geometry ; and indeed it would appear to have 
contained fully half of the propositions subsequently incorporated 
in Theodosius's Sphaerica. This early sphaeric may have origin- 
ated with Eudoxus and his school or may have been older still. 
Its object was purely astronomical ; it did not deal with the geometry 
of the sphere as such, still less did it contain anything of the nature 
of spherical trigonometry (this deficiency was afterwards made good 
by Menelaus's Sphaerica) ; it was designed expressly for such pur- 
poses as fixing the sequence of the times of rising and setting of 
different heavenly bodies, comparing the durations of the risings 
and settings of particular constellations, comparing the apparent 
speeds of the motion of the heavenly bodies at different points 
in their daily revolution, and so on. 4 Perhaps it may best be 

1 A. A. Bjornbo, Studien iiber Menelaos' Spharik. Beitrage zur Geschichte 
der Spharik und Trigonometrie der Griechen (in Abhandlungen zur Geschichte 
der mathematischen Wissenschaften, Heft xiv, 1902), pp. 4, 51, 55. 

2 Bjornbo, op. cit., p. 51. 

8 On the question of Theodosius's date we know little except that he was 
before Menelaus's time. Menelaus made observations in the first year of 
Trajan's reign (a.d. 98) ; and Theodosius, probably of Bithynia, lived before our 
era. Vitruvius (first century B. c.) mentions (ix. 8) a Theodosius who invented 
a sun-dial for all climates, and he may have been contemporary with Hipparchus 
or a little earlier (Tannery, Recherches sur Vhistoire de Ρ astronomie ancienne, 
PP• 36, 37 5 Bjornbo, op. cit., pp. 64, 65). 

4 The sort of thing may be illustrated by the following enunciations of 
propositions : 

Autolycus, On the moving sphere, 9. ' If in a sphere a great circle oblique to 
the axis defines the visible and the invisible (halves) of the sphere [the great 
circle is of course the horizon], then of those points which rise at the same time 



CH. π HISTORY OF TEXT AND EDITIONS 319 

described as the theoretical equivalent of a material sphere or 
combination of spheres (such as are said to have been constructed 
by many astronomers from Anaximander onwards) which should 
exactly simulate the motions of the heavenly bodies and visualize 
the order, &c, of the phenomena as they occur. The special 
necessity for theoretical works of this kind was of course due to the 
obliquity, with reference to the circle of the equator, of (1) the 
horizon at any point of the earth's surface, and (2) the plane of 
the ecliptic in which the independent motions of the sun, moon, 
and planets were supposed to take place. 

We may assume that this mathematical side of astronomy began 
to be studied very early. We know that Oenopides studied certain 
geometrical propositions with a view to their application to astro- 
nomy; and, whether he brought his knowledge of the zodiac and its 
twelve signs from Egypt or not, he was apparently the first to state 
the theory of the oblique movement of the sun. 1 The application of 
mathematics to astronomy may therefore have begun with Oeno- 
pides ; but it had evidently made progress by the time of Archytas, 
Eudoxus's teacher, for Archytas expresses himself, at the beginning 
of a work On Mathematics, thus : 

* The mathematicians seem to me to have arrived at correct con- 
clusions, and it is not therefore surprising that they have a true 
conception of the nature of each individual thing ; for, having 
reached such correct conclusions as regards the nature of the whole 
universe, they were bound to see in its true light the nature of 

those which are nearer the visible pole set later, and of those which set at the 
same time those which are nearer to the visible pole rise earlier.' 

Euclid, Phaenomena, 8. ' The signs of the zodiac rise and set in unequal 
segments of the horizon, those on the equator in the greatest, those next to them 
in the next smaller, those on the tropic circles in the smallest, and those equi- 
distant from the equator in equal segments.' 

Theodosius, Sphaerica, iii. 6. ' If the pole of the parallel circles be on the cir- 
cumference of a great circle and this great circle be cut at right angles by two 
great circles, one of which is one of the parallel circles [i. e. the equator], while the 
other is oblique to the parallel circles [say the ecliptic] ; if then from the oblique 
circle equal arcs be cut off adjacent to one another and on the same side of the 
greatest of the parallel circles [the equator] ; and if through the points so deter- 
mined and the pole great circles be drawn ; the arcs which they will intercept 
between them on the greatest of the parallel circles will be unequal, and the 
intercept which is nearer to the original great circle will always be greater than 
that which is more remote from it.' 

1 Tannery, op. cit., p. 33. 



32o TREATISE ON SIZES AND DISTANCES part π 

particular things as well. Thus they have handed down to us clear 
knowledge about the speed of the stars ^ their risings and settings, 
and about geometry, arithmetic, and sphaeric, and last, not least, 
about music ; for all these branches of knowledge seem to be 
sisters.' 2 

We must suppose, then, that Theodosius's compilation (long- 
winded and dull as it is) simply superseded the earlier text -books 
on Sphaeric, which accordingly fell into disuse and so were lost, 
just as the same fate befell the works of the great Hipparchus as the 
result of their being superseded by Ptolemy's Syntaxis. 

Why Euclid's Optics was included in the Little Astronomy is 
not clear. It was a sort of elementary theory of perspective and 
may have been intended to fore-arm students against the propoun- 
ded of paradoxes such as that of the Epicureans, who alleged that 
the heavenly bodies must be of the size which they appear ; it would 
also serve to justify the assumption of circular movement on the 
part of the stars about the earth as centre. 3 

It was a fortunate circumstance that Aristarchus's treatise found 
a place in the collection ; for presumably we owe it to this fact that 
the work has survived, while so many more have perished. 

Whether Aristarchus had any predecessors in the mathematical 
calculation of relative sizes and distances cannot be stated for 
certain. We hear, indeed, of a book by Philippus of Opus (the 
editor of the Laws of Plato and the author of the Epinomis) 
entitled On the size of the sun, the moon, and the earth, which is 
mentioned by Suidas directly after another work, On the eclipse of 
the moon, attributed to the same author ; but we know nothing of 
the contents of these treatises. 

Like the other books included in the Little Astronomy, our 
treatise passed to Arabia and took its place among the Arabian 
1 middle ' or i intermediate books ', as they were called. It was 
translated into Arabic by Qusta b. Luqa al-Ba'labakki (died about 
912), who was also the translator of the three works of Theodosius, 

1 In connexion with the remark that the mathematicians had investigated the 
speed of the stars, it is perhaps worth while to recall that Eudoxus's great theory 
of concentric spheres was set out in a book On Speeds, nepl ταχών (Simplicius on 
De caelo, p. 494. 12, Heib.). 

2 Porphyry, In Ptolem. Harm., p. 236; Nicomachus, Introd. Arithm. i. 3. 4, 
pp. 6. 17 - 7. 2 ; Vorsokratiker, i 2 , p. 258. 4-12. 

3 Tannery, op. cit., p. 36. 



ch.ii HISTORY OF TEXT AND EDITIONS 321 

Autolycus's On risings and settings, and Hypsicles' Αναφορικός. 1 
A recension of it, as of all the books contained in the Little 
Astronomy, including the Sphaerica of Menelaus, which had been 
translated by Ishaq b. Hunain, was made by Nasiraddin at-Tusi, 2 
famous as the editor of Euclid and for an attempt to prove Euclid's 
Parallel-Postulate. There are MSS. of this collection, including 
of course Aristarchus, in the India Office (743, 744) and in the 
Bodleian Library (Nicoll and Pusey, i. 875, i. 895, and ii. 279). 

The first published edition of Aristarchus 's treatise was a Latin 
translation by George Valla, included in a volume which appeared 
first in 1488 ('per Anton, de Strata') and again in 1498 ('per 
Simonen Papiensem dictum Bevilaquam '). 3 

It next appeared in a Latin translation by that admirable and 
indefatigable translator Commandinus, under the title : 

A r is fare hi de magnitudinibus et distantiis so/is et lunae liber 
cum Pappi A lexandrini explicationibus qtiibusdam a Federico 
Commandino Urbinate in latinum conversus et commentariis 
illustratus. Pisauri, 1572. 

Commandinus complains of the state of the text, which made the 
task of translation difficult, but he does not mention Valla's earlier 
translation and was presumably not acquainted with it. 

The honour of bringing out the editio princeps of the Greek 
text belongs to John Wallis. The title-page is as follows : 

1 Suter, Die Mathematiker und Astronomen der Araber tend ihre Werke 
(Abh. zur Gesch. d. math. Wissenschaften, x. Heft, 1900), p. 41. 
3 Suter, p. 152. 
3 Fabricius, Bibliotheca Graeca, iv. 19, Harles. 



1410 



ΑΡΙΣΤΑΡΧΟΥ 2ΑΜΙΟΪ 

Πβ&Ι μαγψίιΐ % ~&!ηφι\$ Ηλί'ν £ 2ελία5«ί, 

Β Ι Β Λ Ι Ο Ν. 

ΠΑΠΠΟΙ ΑΛΕΗΑΝΔΡΕΠΣ 

Τ«•?Σία/Λ^>ίίςΒΐΒΛΙΟΪ Β' 

ARISTARCHI SAMII 

De Magnitudinibus & Diftantiis Solis & Lunae, 

LIBER. 

Nunc primum Greece editus cum Federici Com- 
mandini verfione Latina, notify, illius & Editoris. 

PAPPI ALEXANDRINI 

Secundi Libri 

Mathematics Collectionis, 

Fragmentum, 

Haitenus Defideratum. 

Ε Codice MS. edidit, Latinum fecit, 

Notifque illufiravit 

JOHANNES WALLIS, S. T. D. Geometric 

Profeflor Savilianus ; & Regalis Societatis 

Londini , Sodalis. 

ο χ ο ν ι M, 

Ε Theatro Sheldoniano, 

itf88. 



ch.ii HISTORY OF TEXT AND EDITIONS 323 

The book was reprinted in the collected edition of Johannis 
Wallis Opera Mathematical 1 693-1 699, vol. iii, pp. 565-94. 

Wallis states in his Preface that he used for the preparation of his 
text (1) a Greek MS. (which he calls B) belonging• to Edward 
Bernard, Savilian Professor of Astronomy, who had copied it from 
the Savile MS., and (2) the Savile MS. itself (S). The Savile MS. 
was copied by Sir Henry Savile himself from another (presumed 
by Wallis to have been one of the Vatican MSS.), and had (as 
appeared from notes in the margin) been collated with a second 
MS. vaguely described as Codex Vetus. Wallis preferred Com- 
mandinus's translation to Valla's, and retained the former version 
intact because it agreed so closely with the Greek MSS. of 
Savile and Bernard that it seemed to have a common source with 
them ; Wallis also incorporated Commandinus's notes along with 
his own. 

Wallis adds that there are two Selden MSS. in the Bodleian 
Library containing A ristarchus's treatise in Arabic, and that Bernard 
had noted in the margin of his MS. (B) anything in the Arabic 
version which seemed of moment, as well as some things from 
Valla's translation. 

In 1 810 there appeared the edition by the Comte de Fortia 
d'Urban, 

Histoire dAristarque de Santos, suivie de la traduction de son 
ouvrage sur les distances du Soleil et de la Lune, de Γ histoire 
de ceux qui ont porte le nom d'Aristarque avani Aristarque 
de Santos, et le commencement de celle des Philosophes qui 
ont par u avant ce meme Aristarque. Par M. de F * * * *. 
Paris, ίδιο. 

There follows, as a separate title-page for the work of Aristarchus, 
Άριστάργου nepl μβγβθών καϊ αποστημάτων ηλίου καϊ σελήνης, 
followed by the Latin equivalent. Pages 2-87 contain the Greek 
text along with Commandinus's Latin translation (altered in places). 
On p. 88 is a note referring to the MSS. used by the editor in pre- 
paring the Greek text of the treatise and the scholia. The scholia 
in Greek and Latin occupy pages 89-199, and are followed by the 
critical notes, which extend from p. 201 to p. 248. Particulars of 
the MSS. used will be found in a later paragraph. 

Υ 2 



324 TREATISE ON SIZES AND DISTANCES part π 

This Greek text of Fortia d'Urban was issued prematurely and 
without any diagrams ; an explanation on the subject is contained 
in the editor's preface to his French translation published thirteen 
years later, 

Traite d'Arisiarque de Samos sur les grandeurs etles distances 
du Soleil et de la Lune, traduit en francais pour la premiere 
fois^par M. le Comle de Fortia d'Urban. Paris, 1823. 

The Preface to this translation, with the omission of an explana- 
tion of the lettering in the figures (which is double, to correspond 
to the Greek text and the Latin and French translations), runs as 
follows : 

4 Le texte de l'ouvrage d'Aristarque de Samos, que j'avais revu 
sur huit manuscrits de la bibliotheque du Roi, et que j'avais fait 
imprimer en France ou il n'avait point encore ete publie, avec des 
scholies absolument inedits, ayant ete mis en vente sans mon 
automation, a paru dune maniere presque ridicule. On y trouve 
citees, a toutes les pages, des planches que j'avais fait graver, mais 
que des circonstances facheuses ont fait disparaitre pendant mon 
sejour en Italie. Je vais tacher d'y suppleer par la publication de 
cette traduction qui sera accompagnee de nouvelles planches ou 
j'ai fait graver les lettres grecques pour ceux qui voudront joindre 
cette traduction au texte . . . Je donnerai d'abord l'ouvrage d'Aris- 
tarque de Samos, tel qu'il nous est parvenu ; je traduirai ensuite les 
scholies, suivant ainsi l'ordre observe pour l'impression du texte 
grec. J'avertis que les demonstrations d'Aristarque s'appuient sur 
la Geometrie d'Euclides, qu'il suppose connue de ses lecteurs. 
Paris, 2 avril 1823.' 

The French translation is a meritorious and useful book. 
There is yet another Greek text, besides those of Wallis and 
Fortia d'Urban, namely 

'Αριστάργ^ον Σαμίου βίβλων nepl μβγζθων και αποστημάτων ήλιου 
καϊ σελήνης, mit kritischen Berichtigungen von Ε. Nizze. 
Stralsund, 1856. 

This text is, however, untrustworthy, not having been prepared 
with sufficient care. It was based on the texts of Wallis and Fortia 
d'Urban without, apparently, any recourse to MSS. 
A German translation also exists, 

Arisiarchus uber die .Grossen und Entfemungen der Sonne 
und des Mondes, ubersetzt und erlatitert von A. Nokk. 
Freiburg i. B., 1854. 



CH. π HISTORY OF TEXT AND EDITIONS 325 

We come now to the MS. authority for our Greek texts. It 
would appear 1 that our treatise is included in five MSS. in the 
Vatican, namely Vat. Gr. 204 (10th cent.), 191 and 203 (13th cent.), 
192 (14th cent.), and 202 (i4th-i5th cent.), and in eight at Paris, 
namely Paris. Gr. 2342 (14th cent.), 2363 (15th cent.), 2364, 2366 
(16th cent.), 2386 (16th cent.), 2472 (14th cent.), 2488 (16th cent.), 
and Suppl. Gr. 12 (16th cent.). There are others at Venice, Mar- 
cian. 301 and 304 (15th cent.); at Milan, Ambros. A 101 sup. (14th 
cent.); at Vienna, Vindobon. Suppl. Gr. 9 (17th cent.); and so on. 

The oldest of all these MSS. and by far the best is the beautiful 
Vaticanus Graecus 204, of the 10th century; indeed it seems to be 
the ultimate source of all the others, and so much superior that the 
others can practically be left out of account. This great MS. is 
described by Menge. 2 Its contents are : fol. i-36 v Theodosius, 
Sphaerica, i, ii, iii; 37 r ~42 v , Autolycus, On the moving sphere ; 
42 v ~58 r , Prolegomena to Euclid's Optics (τα προ των ΕύκλζίΒου 
οπτικών) ; 3 58 Γ -7°Λ Euclid's Phaenomena ; 7^-82 ν , Theodosius, On 
habitations; 83 Γ ~95 Γ , Theodosius, On nights and days; 95 r -io8 v , 
Theodosius, On days and nights \ ii ; ιοδ ν - 1 1 7 V , Aristarchus, On the 
sizes and distances of the sun and moon ; 1 1 8 Γ - 1 32 r , Autolycus, On 
risings and settings, i, ii ; I32 v -i34 v , Hypsicles, 'Αναφορικός ; i35 r - 
1 43 v , Euclid's Caloplrica ; 1 44 r , figures to the Catoptric a ; 1 44 v blank ; 
I45 r -i72 r , Eutocius, Commentary on Books I-III of Apollonius's 
Conies; i72 v -i94 v , Euclid's Data ; i95 r -i97 r , Marinus, Commen- 
tary 07t Eticlid's Data ; 1 9δ Γ -2θ5 ν , Scholia to Euclid's Elements. 

The MS. is of parchment, incomplete at the end, and the 206 
leaves are preceded by three more, the first of which is empty, the 
second has a πίναξ, and the third, a sheet of paper fastened in later, 
contains a Latin index. The first two leaves, containing the begin- 
ning of Theodosius's Sphaerica, are written by a later hand who 

1 I have collected these particulars, except as regards three of the MSS. used 
by Fortia d'Urban, from the introductions to Heiberg's editions of Euclid and 
Apollonius in Greek, the same scholar's Literargeschichtliche Studien iiber 
Euklidy 1882, and Om Scholierne til Euklids Elementer, 1888, and from the 
introductions to one or two other Greek mathematical texts. 

2 Addendum to a review of Hultsch's Autolycus, Neue Jahrbiicher fur 
Philologie, 1886, pp. 183, 184. 

3 Fol. 42 v -58 r contain Theon's recension of Euclid's Optics, with a preface 
which was apparently written by some pupil of Theon's. It is to this preface 
that the title refers. 



$26 TREATISE ON SIZES AND DISTANCES part π 

has cleverly imitated the handwriting of the rest of the MS., which 
is by one hand. The figures, drawn in red, are clear and adequate. 1 
Many things in the text are struck out, erased, and changed by 
different hands. The MS. is rich in old and new scholia. It has 
on it the stamp of the Bibliotheque Nationale, having been, like the 
famous Peyrard MS. of Euclid (Vat. Gr. 190), among the MSS. 
which were taken to Paris in 1 808 and restored to the Vatican after 
the Congress of Vienna. 

In settling a text to translate from, I have mainly relied on a 
photograph of Vat. Gr. 204 together with Wallis's text, though 
I have had Nizze's text by me and have also consulted Fortia 
d'Urban's edition of 1810. The occasional references to the 
Paris MSS. in my critical notes are taken from Fortia d'Urban. 2 

It is not clear from which of the Vatican MSS. Savile copied his 
own (Wallis's S) ; it cannot, however, have been Vat. Gr. 204, because 
[a) nearly all the words and sentences which Wallis supplied, on the 
basis of Commandinuss translation, in order to fill up gaps in his 
two MSS., are actually found (either exactly or with no more 
variation than would naturally be expected between a re-translation 
into Greek and the original Greek text) in 204, and [δ) a scholium 
from S added by Wallis at the end of Prop. 7 does not appear in 
204. Fortia d'Urban suggests, as a possibility, that the MS. of 
which Wallis had a copy was Paris. 2366, but it seems clear that it 
cannot have been any of the Paris MSS., and therefore was pre- 
sumably (as Wallis thought) one of those in the Vatican. 3 There 

1 The words used by Menge are * klar und genau ', but I think the figures can 
hardly be called i accurate ' or ' exact '. 

2 In Fortia d'Urban's critical notes there are several references to the 
reading of a MS. which he quotes as 2483. But Paris. Gr. 2483 is not included 
in his list of the MSS. of Aristarchus used by him; and it appears to contain, 
not Aristarchus, but Nicomachus's Introductio arithmetica with scholia (Omont, 
Inventaire sommaire des manuscrits grecs de la Bibliotheque Nationale, ii). 
It would seem, from internal evidence, that the references should be to Paris. 
Gr. 2472, not 2483, in these cases. 

3 Fortia d'Urban observes that Paris. 2366 alone omits a sentence in Prop. I 
(πολλω apa η Β Γ της Β Α ελάσσων εστίν η με' μέρος) which Wallis likewise Omits, 
whereas Paris. 2342, 2364, 2488 and Commandinus all have it ; hence he thinks 
that Wallis's MS. may have been a copy of Paris. 2366. But, on the other hand, 
a sentence in Prop. 13 which is absent from Wallis's text (και ή Β Ν εφάπτεται . . . 
λαμπρόν) is, according to Fortia d'Urban, found in all the Paris MSS. except 
2342 ; presumably therefore Paris. 2366 has it. These two cases create a strong 
presumption that Wallis's MS. was not a copy of any of the Paris MSS. 



ch.ii HISTORY OF TEXT AND EDITIONS 327 

is apparently no clue to the identity of the 4 Codex Vetus ' with 
which S was collated. 

We are better informed as to the MSS. used by Fortia d'Urban. 
He tells us, in the note on p. 88 of the edition of 18 10, that they 
were Codd. Paris. 2342, 2363, 2364, 2366, 2386, 2472, and 2488, 
and one Vatican MS. The particular Vatican MS. had, he observes, 
just been brought to Paris ; it was therefore presumably Vat. Gr. 
204. He does not, however, seem to have collated the latter MS. 
with sufficient care ; for example, he says that some words * in the 
' setting-out ' of Prop. 3 and a whole sentence 2 occurring later in 
the proposition are wanting in the MS., though, as a matter of fact, 
they are there in full ; when, therefore, on the occasion of the first 
of these supposed omissions, he says that the Vatican MS. does not 
seem to him in any way superior to ' our own ', he is apparently 
allowing his patriotism to get the better of his judgement. For the 
scholia he says that he relied mostly upon Paris. 2342 and 2488 ; 
but the scholia in Vat. Gr. 204 seem to correspond exactly. He 
does not seem to have found in any of his eight MSS. the particular 
scholium to Prop. 7 taken by Wallis from S ; for, while he gives it 
in his French translation, he says it comes, through Wallis, from S. 

1 σελήνης Se κίντρον, όταν 6 περιλαμβάνων κώνος 

2 και δκλόντι, ως η Β Τ 1 προς την Γ Λ, όντως ή Β Δ προς την ΔΟ. 



Ill 
CONTENT OF THE TREATISE 

The style of Aristarchus is thoroughly classical, as befits an able 
geometer intermediate in date between Euclid and Archimedes, 
and his demonstrations are worked out with the same rigour as 
those of his predecessor and successor. The propositions of 
Euclid's Elements are, of course, taken for granted, but other things 
are tacitly assumed which go beyond what we find in Euclid. 
Thus the transformations of ratios defined in Eucl. V and indicated 
by the terms inversely, alternately, componendo, convertendo, &c, 
are regularly and naturally used in dealing with unequal ratios, 
whereas in Euclid they are only used in proportions, i. e. cases of 
equality of ratios. But the propositions of Aristarchus are also of 
particular mathematical interest because the ratios of the sizes and 
distances which have to be calculated are really trigonometrical 
ratios, sines, cosines, &c, although at the time of Aristarchus trigono- 
metry had not been invented, while no reasonably close approxima- 
tion to the value of π, the ratio of the circumference of a circle to its 
diameter, had been made (it was Archimedes who first obtained the 
value 22 /η). Exact calculation of the trigonometrical ratios being 
therefore impossible for Aristarchus, he set himself to find upper 
and lower limits for them, and he succeeded in locating those which 
emerge in his propositions within tolerably narrow limits, though 
not always the narrowest within which it would have been possible, 
even for him, to confine them. In this species of approximation to 
trigonometry he tacitly assumes propositions comparing the ratio 
between a greater and a less angle in a figure with the ratio 
between two straight lines in the figure, propositions which are 



CONTENT OF THE TREATISE 329 

formally proved by Ptolemy at the beginning of his Syntaxis. 
Here, again, we have a proof that text-books containing such 
propositions existed before Aristarchus's time, and probably much 
earlier, although they have not survived. 

The formal assumptions of Aristarchus and 
their effect. 

One of the assumptions or hypotheses at the beginning of the 
treatise, the grossly excessive estimate of 2 for the apparent 
angular diameter of the moon, has already been discussed (pp. 311, 
312 above). We proceed to Hypotheses 4 and 5, giving values 
for a certain ratio and a certain other angle respectively. 

In Hypothesis 5, Aristarchus takes the diameter of the earth's 
shadow (at the place where the moon passes through it at the time 
of an eclipse) to be twice that of the moon. The figure 2 for this 
ratio was presumably based on the observed length of the longest 
eclipses on record. 1 Hipparchus, as we learn from Ptolemy, 2 made 
the ratio 2\ for the time when the moon is at its mean distance in 
the conjunctions ; Ptolemy chose the time when the moon is at its 
greatest distance, and made the ratio insensibly less than 2| (a 
little too large). 3 

Tannery 4 shows in an interesting way the connexion between 
(1) the estimate (Hypothesis 4) that the angular distance between 
the sun and moon viewed from the earth at the time when the 
moon appears halved is 87 , the complement of 3 , (2) the estimate 
(Hypothesis 5) of 2 for the ratio of the diameter of the earth's 
shadow to that of the moon, and (3) the ratio (greater than 18 to 1 
and less than 20 to 1) of the diameter of the sun to the diameter of 
the moon as obtained in Props. 7 and 9 of our treatise. 

The diagram overleaf (Fig. 14) will serve to indicate very roughly 
the relative positions of the sun, the earth, and the moon at the 
moment during a lunar eclipse when the moon is in the middle of 
the earth's shadow. 

1 Tannery, Recherches sur Vhistoire de V astronomic ancienne, p. 225. 

2 Ptolemy, Syntaxis, iv. 9, p. 327. 3-4, Heib. 

3 Ibid., v. 14, p. 421. 12-13. 

* Tannery in Mimoires de la Societe des sciences physiques et naturelles de 
Bordeaux, 2 Θ serie, v, 1883, pp. 241-3 ; Memoires scietitifiques, ed. Heiberg and 
Zeuthen, i, 1912, pp. 376-9. 



33o TREATISE ON SIZES AND DISTANCES partii 







s\ / 

Λ | / 








r 


ΓλΜοοπ. 






5 




Ϊ 


^ 




r^ 


"Η 


Sura. 


Earth. 




V 


3^^ 





Fig. 14. 



Let kS be the radius of the sun's orbit, 
L moon's orbit, 

s the radius of the sun, 
/ ... moon, 

/ ... earth, 

D the distance from the centre of the earth to the vertex of 
the cone of the earth's shadow, 
and d the radius of the earth's shadow at the distance of the moon. 
Then we have, approximately, by similar triangles, 
D _/_ d_D-L, 



s-t 



D 



LI d 

whence, if we suppose that -^ = - , and put n—^ y we easily 



derive 
and 






D 



L t-d 



I 
1 — n- 



■ (0 



Now, since eclipses of the sun occur through the interposition of 
the moon, S > L, so that s > L The ancients knew, too, that the 
sun is larger than the earth, so that s > t. It follows from (1) that 

j > n, so that the moon is smaller than the earth. 

Now suppose δ to be the angle subtended at the centre of the sun 
by the distance between the moon and the earth at the time when 
the moon appears halved, i.e. when the earth, sun, and moon form 



CH. Ill 



CONTENT OF THE TREATISE 



33 1 



a right-angled triangle with its right angle at the centre of the 
moon. 

Let x = — — - — — 



L ~ I sin δ ' 
We have then from (i), substituting s/x for /, 



tx t s x+ 1 

- + - = n+i , or - = — — 

^ ^ / n+ ι 



and, substituting be for s, we have 







Fig. 15. 



Now if λ: (= 7) is taken at 19, Aristarchus's mean value, and 



/ 

72 = 2, these formulae give 
»? <y 20 ^ 5 

/ =I9 ' ^ = 7 " 6f 



- 7 =2-85, 5 = sin- 1 — = 3°i , i ,/ . 



Tannery's object is to prove that the method of our treatise was 
not invented by Aristarchus but by Eudoxus. We know in the 
first place, from Aristotle, that by the middle of the fourth century 
mathematical speculations on the sizes and distances of the sun and 
moon had already begun. Aristotle 2 says : 

1 Besides, if the facts as shown in the theorems of astronomy are 
correct, and the size of the sun is greater than that of the earth, 
while the distance of the stars from the earth is many times greater 
than the distance of the sun, just as the distance of the sun from the 
earth is many times greater than that of the moon, the cone marking 
the convergence of the sun's rays (after passing the earth) will have 
its vertex not far from the earth, and the earth's shadow, which we 
call night, will therefore not reach the stars, but all the stars will 
necessarily be in the view of the sun, and none of them will be 
blocked out by the earth.' 

1 Arist. Mefeorologica i i. 8, 345 b 1-9. 



$$% TREATISE ON SIZES AND DISTANCES part Ή 

Now Eudoxus was the first person to develop scientifically the 
hypothesis that the sun and moon remain at a constant distance 
from the earth respectively, and this is the hypothesis of Aristar- 
chus. Further, we are told by Archimedes that Eudoxus had 
estimated the ratio of the sun's diameter to that of the moon at 
9:1, Phidias, Archimedes' father, at 12:1, and Aristarchus at a 
figure between 18:1 and 20:1. Accordingly, on the assumption 
that Eudoxus and Phidias took η = 2 in the above formulae, as 
Aristarchus did, we can make out the following table : 





^ 


^ 




/ 


/ 


Eudoxus 


9 


3'3 


Phidias 


12 


4'3 


Aristarchus 


19 

(mean) 


6-6 



27 
276923 

2.85 



(calculated value) 



6° 22' φ' 
4° 46' 49' 



Hence, says Tannery, while Aristarchus took 3 as the value of 

#, Eudoxus probably took 6° or §- of a sign of the zodiac, and 

Phidias 5 or J of a sign. ' I cannot believe that these values were 

deduced from direct observations of the angular distance. The 

necessary instruments were in all probability not in existence in the 

fourth century. But Eudoxus could, on the day of the dichotomy, 

mark the positions of the sun and the moon in the zodiac, and try 

to observe at what hour the dichotomy took place. The evaluations 

involve an error of about twelve hours for Eudoxus, ten for Phidias, 

and six for Aristarchus. It seems that all of them sought upper 

limits for 8. It will be noticed that the value of δ especially affects 

the values of the ratios s/l, s/i ; the ratio /// on the contrary depends 

mostly on the value of η? l Seeing, however, that the only figures 

in the above tables which are actually attested are the three in the 

first column, the 3 of Aristarchus, and the results obtained by 

Aristarchus on the basis of his assumptions, it seems a highly 

speculative hypothesis to suppose that Eudoxus started with 6°, 

and Phidias with 5 , as Aristarchus did with 3 , and then deduced 

the ratio of the diameter of the sun to that of the moon by precisely 

Aristarchus's method. 

1 Tannery, Memoires de la Societe des sciences fihys. et nat. de Bordeaux, 
2 Θ s^rie, v, 1883, pp. 243-4 > Memoires scientifiques, ed. Heiberg and Zeuthen, 
h P- 379- 



ch.iii CONTENT OF THE TREATISE 333 

Trigonometrical equivalents. 

Besides the formal Assumptions laid down at the beginning- of 
the treatise, there lie at the root of Aristarchus's reasoning certain 
propositions assumed without proof, presumably because they were 
generally known to mathematicians of the day. The most general 
of these propositions are the equivalent of the statements that — 

If α is what we call the circular measure of an angle, and oc is 
less than § 77-, then 

(1) The ratio sin α/α decreases as α increases from to |ττ, 

but (2) the ratio tan ol/ol increases as α increases from to \ π. 

Tannery l took pains to set out the trigonometrical equivalents 
of the particular results obtained by Aristarchus in the several 
propositions. 

If we bear in mind that 

. 7Γ 7Γ 

sin- = tan - = 1, 

2 4 

sin| = |, 

7Γ I 

tan-~ = 



8 V2 + 1 ' 

and if we substitute for V '2 the approximate value § which is 
assumed by Aristarchus, we can deduce the following inequalities : 

( 1 ) If m > 1 , sin — > — , 
v ' 2m m 

, \ 7Γ . ,π π \ ^ nt—\ 

or (2) cos = sin ( - — ) > , 

v J 2m v 2 2m ; m 

/ \ rr . 7Γ 7Γ 2 

(3) If m > 2, sin < tan < — , 

2m 2m m 

(4) Vim > 3, sin π > -^-, 
J 2 m 2 m 

ι \ τλ . 7Γ 7Γ Κ 

(κ) It m > 4, sin < tan < -^- . 

2 m 2m %m 

1 Tannery, Memoires de la Soc. des sciences fihys. et nat. de B07 deaux, 2 e serie, 
v, 1883, pp. 244 sq. ; Memoi?'es scie?itifiques y i, pp. 380 sqq. 



334 TREATISE ON SIZES AND DISTANCES partii 

The narrowest limits for sin obtained by means of these 

inequalities are 

(6) S- > sin — > -1- , 

v 3#Z 2»Z 2W 

whereas, if Aristarchus had known the approximate value - 2 T 2 - for π, 
he could have obtained the closer upper limit 

π ii 

sin — < 

2m im 

Now, for example, in Prop. 7, Aristarchus has to find limits for 

7Γ 

sin 3 , that is to say sin -p- ; thus m — 30, and the formula (6) 

above gives his result 

1 1 

-^ > sin 3 > — . 

18 " 20 

In Prop. 4 Aristarchus proves the negligibility of the maximum 
angle (e) subtended at the centre of the earth by a certain arc (a) on 
the surface of the moon subtended at the centre of the moon by an 
angle equal to half the apparent angular diameter of the moon. 
From the figure of the proposition it is easy to see that, taking the 
radius of the moon to be unity, 





e 

tan - = 

2 


. α 
sin α sin - 








a 
1— sin α cos - 

2 




For, if Μ be the foot of the perpendicular from Η 


on AB, 




e HM 

tan ' = ΤΓ7 = 

2 MA 


HM 


BHsm- 

2 






~ AB-MB' 


AB-BH cos- 
2 








BD sin - 

2 


. α 

sin α sin - 

2 








AB-BD cos- 1 

2 


— sin ol cos 


(X 

2 



This would give, for α = i°, e = o° 1' 3". 



ch.iii CONTENT OF THE TREATISE 335 

What Aristarchus in fact does is to prove that 

. € 

e Sm 2. BH BG BG . sin α 

- < 1, e. < -mr < -fth or -=r-; ^-^ 1• e. < 



α . α HA GA Β A -BG ' * 1 -sin α 

sin - 

2 

Now, if α = π/2 m (m > 4), formula (5) above would give 

2m * 5 j-r 2€ ^ x 0/0// 
< — 2 — and, if m = go, — < ; €<o 1 8 : 

7Γ 3^-5 ^ 477° 

but Aristarchus is content with the equivalent of using formula (3) 
which gives 

2 6 I 2 I ο / // 

— < — i.e. < — p- , or € < ο ι 22 . 

π m m — 2 3960 

In Prop. 1 1 Aristarchus uses the equivalent of formulae (3) and 

(4), proving that 

1 . ο 1 
— > sin 1 > 7- • 

45 6o 

Prop. 1 2 is the equivalent of using formula (2) to prove that 

ο 8 9 
I >COS I > - z • 

90 

From formula (2) we deduce 

„ π m 2 — 2M + i m — 2 

cos 2 > 5 > ; 

2 m m l m 

and, for m = 90, this gives the equivalent of the first part of 
Prop. 1 3, namely „ 44 

r J ' J I >COS 2 I > — . 

45 
In Prop. 14 Aristarchus determines a lower limit for the ratio 
L]c, where L is the radius of the moon's orbit and c the distance of 
the centre of the moon from the centre of the circle of the shadow 
at the middle of an eclipse. The arithmetical value of the limit 
depends of course on the particular assumptions which he makes 
as to the angles subtended at the centre of the earth by the 
diameter of the moon and by the diameter of the circle of the 
shadow. If these angles be 2 a, 2 γ respectively, we see from 
the figure of Prop. 14 that 

BR = Β Μ cos α = L cos 2 a, BS = L cos a cos γ, RC — L sin 2 a. 



33 6 TREATISE ON SIZES AND DISTANCES part π 

Therefore SR : RC = L cos α (cos oc — cos y) : Ζ sin 2 a, 
and CR \ CS = L sin 2 α : Ζ (sin 2 α + cos 2 α — cos oc cos γ) 

= L sin 2 α : Ζ (i — cos α cos y). 

Now BC:CR= (BC: CM) χ (CM: CR) 

= (i :sina) χ (ι :sin oc) 
= ι : sin 2 a. 

Therefore, ex aequali, 

BC : CS = L:L(i —cos oc cos y), 

or Ζ :c — ι : (i— cos a cosy) 

= ι : (sin 2 oc + cos 2 oc — cos oc cos y ) 
> ι : { sin 2 a + cos 2 a ( ι — cos y) } . 

If y = 2 a, as assumed by Aristarchus, this becomes 
L : 0(1 : sin 2 a) . {i : (i +2 cos 2 a)}. 

The corresponding inequality obtained by Aristarchus, who 
assumes that α = i°, is 

Z:^>(45:i) 2 .(i: 3 ) 
> 675:1. 

The generalized trigonometrical equivalent of Prop. 15 is more 
complicated and need not be given here. Tannery has an inter- 
esting remark, which was however anticipated by Fortia d'Urban, 1 
upon one of the arithmetical results obtained by Aristarchus in that 
proposition. If y be the ratio of the sun's radius to the earth's 
radius, his result is 

y > 71755875 



y- 1 6 i7355°° 
He replaces this value by ff, merely remarking that ( 71755875 
has to 61735500 a ratio greater than that which 43 has to 37 '. It 
is difficult, says Tannery, not to see in || the expression 1 +| +1 , 

■q 

which suggests that |f was obtained by developing Jf?|||^ or 
ill fi- as a continued fraction. ' We have here an important proof 
of the employment by the ancients of a method of calculation, the 
theory of which unquestionably belongs to the moderns, but the 
first applications of which are too simple not to have originated in 
very remote times.' 

1 Fortia d'Urban, Traite d'Aristarqae de Santos, 1823, p. 86, note. 



IV 



LATER IMPROVEMENTS ON ARISTARCHUS'S 

CALCULATIONS 

While it would not be consistent with the plan of this work to 
carry the history of Greek astronomy beyond Aristarchus, it will be 
proper, I think, to conclude this introduction with a few particulars 
of the improvements which later Greek astronomers made upon 
Aristarchus 's estimates of sizes and distances. 

We have already spoken of Aristarchus 's assumption of 87 as the 
angle subtended at the centre of the earth by the line joining 
the centres of the sun and moon at the time when the moon 
appears halved. The true value of this angle is 89 50', so 
that Aristarchus's estimate was decidedly inaccurate ; no direct 
estimate of the angle seems to have been made by his successors, 
Aristarchus himself, as we have seen, afterwards corrected to f ° the 
estimate of 2° for the apparent angular diameter of the sun and 
moon alike. His assumption of 2 as the ratio of the diameter of 
the circle of the earth's shadow to the diameter of the moon was 
improved upon by Hipparchus and Ptolemy. Hipparchus made 
it 2\ at the moon's mean distance at the conjunctions; 1 Ptolemy 
made it at the moon's greatest distance l inappreciably less than 2f \ 2 

Coming now to estimates of absolute and relative sizes and 
distances, we find some data in Archimedes ; 3 according to him 
Eudoxus had declared the diameter of the sun to be nine times the 
diameter of the moon, and Phidias (Archimedes' father) twelve times ; 
most astronomers, he says, agreed that the earth is greater than the 
moon, and ' some have tried to prove that the circumference of the 
earth is about 300,000 stades and not greater \ It seems probable 
that it was Dicaearchus who (about 300 B.C.) arrived at this value, 4 

1 Ptolemy, Syntaxis, iv. 9, vol. i, p. 327. 3-4, Heib. 

2 Ibid. v. 14, vol. i, p. 421. 12-14, Heib. 

3 Archimedes, Sand-reckoner (Archimedis opera, ed. Heib., vol. ii, p. 246 sqq ) : 
The Works of Archimedes, pp. 222, 223. 

4 Berger, Geschichte der wissenschaftlichen Erdkunde der Griechen y 
pp. 370 sqq. 

1410 Ζ 



338 LATER IMPROVEMENTS ON part π 

and that it was obtained by taking 24 (i/i5th of the whole meri- 
dian circle) as the difference of latitude between Syene and Lysi- 
machia (on the same meridian), and 20,000 stades as the actual 
distance between the two places. 1 Archimedes' own estimates are 
scarcely estimates at all ; they are intentionally exaggerated, as, his 
object being to measure the number of grains of sand that would 
fill the universe, he adopts what he considers maximum values in 
order to be on the safe side. Thus he says that, whereas Aristar- 
chus tried to prove that the ratio of the diameter of the sun to that 
of the moon is between 18:1 and 20 : 1 , he himself will take the 
ratio to be 30 : 1 and not greater, in order that his thesis may be 
proved ' beyond all cavil ' ; in the case of the earth he actually 
multiplies the estimate of the perimeter by 10, making it 3,000,000 
instead of 300,000 stades. 

Before passing on to later writers, it will be convenient to re- 
capitulate Aristarchus's figures ; and for brevity I shall use the 
letters by which Tannery denotes the various distances and radii, 
namely S for the distance of the centre of the sun, L for the 
distance of the centre of the moon, from the centre of the earth, 
and s, /, / for the radii of the sun, moon, and earth respectively. 
Aristarchus's figures then are as follows : 

L/%1 > 22\ but < 30 (Prop. 11). 

S/L > 18 but < 20 (Prop. 7). 

2s/2t or s/t > 6§ but < η\ (Prop. 15). 
2i/7,torl/t > |f but < τ \% (Prop. 17). 

We may with Hultsch, 2 for convenience of comparison with other 
calculations, take figures approximating to the mean between the 
higher and lower limits ; and it will be convenient to express 
the various diameters and distances in terms of the diameter of the 
earth. We may say then, roughly, that 

*//*' = * e * = A; 

2s/2i— 6J; 
Z/2/ = 26i; 
S/L = 19: 

1 Cf. Cleomedes, De 7notu circularly i. 8, p. 78, Ziegler. 

2 Hultsch, Poseidonios iiber die Grosse und Entfernung der Sonne, 
1897, p. 5. 



ch.iv ARISTARCHUS'S CALCULATIONS 339 

whence 

L/U = ψ^ = 9 _ % say 9 i . 

6/2/ = \ 8 o 9 - . 19 = 179»», say 180. 

We are not told what size Aristarchus attributed to the earth, but 
presumably, like Archimedes, he would have accepted Dicaearchus's 
estimate of 300,000 stades for its circumference. 

Eratosthenes (born about eleven years after Archimedes, say 
276 B.C.) is famous for a measurement of the earth based on 
scientific principles. He found that at noon at the summer solstice 
the sun threw no shadow at Syene, while at the same hour at 
Alexandria (which he took to be on the same meridian) it made 
the gnomon in the scaphe cast a shadow showing an angle equal 
to one-fiftieth of the whole meridian circle ; assuming, further, that 
the sun's rays at Syene and Alexandria are parallel in direction, 
and that the known distance from Syene to Alexandria is 5,000 
stades (doubtless taken as a round figure), Eratosthenes arrived by 
an easy geometrical proof at 50 times 5,000 or 250,000 stades as 
the circumference of the earth. This is the figure given by 
Cleomedes ; * but Strabo quite definitely says that, according to 
Eratosthenes, the circumference is 252,000 stades, 2 and this is the 
figure which is most generally quoted in antiquity. The reason 
for the discrepancy has been the subject of a good deal of discus- 
sion ; 3 it is difficult, in view of Cleomedes' circumstantial account, 
to suppose that 252,000 was the original figure arrived at by 
Eratosthenes ; it seems more likely that Eratosthenes himself cor- 
rected 250,000 to 252,000 for some reason, perhaps in order to get 
a figure divisible by 60 and, incidentally, a round number (700) of 
stades for one degree. There is some question as to the length of 
the particular stade used by Eratosthenes, but, if Pliny is right in 
saying that Eratosthenes made 40 stades equal to the Egyptian 
σχοΐνος* then, taking the σχοΐνος at 12,000 Royal cubits of 0-525 
metres, 5 we get 300 such cubits, or 157-5 metres, as the length ot 
the stade, which is thus equal to 516-73 feet. The circumference 
of the earth, being 252,000 times this length, works out to about 

1 Cleomedes, De motu circulari, i. 10, especially p. 100. 15-23, ed. Ziegler. 

2 Strabo, ii. 5. 7, p. 113 Cas. s Berger, op. cit., pp. 410, 411. 
4 Pliny, N.U.xu. c. 13, §53. 

6 Hultsch, Griechische u. romische Metrologie (Berlin, 1882), p. 364. Cf. 
Tannery, Recherches sur Vhistoire de Vastronomie ancienne, pp. 109, no. 

Ζ 2 



34θ LATER IMPROVEMENTS ON partii 

24,662 miles, and the diameter of the earth on this basis is about 
7,850 miles, only 50 miles shorter than the true polar diameter, 
a surprisingly close approximation, however much it owes to happy 
accidents in the calculation. 1 

We have no trustworthy information as to evaluations by Erato- 
sthenes of other dimensions and distances. The Doxografihi, it is 
true, say that Eratosthenes made L, the distance of the moon from 
the earth, to be 78 myriads of stades, and S, the distance of the 
sun, to be 80,400 myriads of stades 2 (the versions of Stobaeus and 
Joannes Lydus admit of 408 myriads of stades as an alternative 
interpretation, but this figure obviously cannot be right). Tannery 3 
considers that none of these figures can be correct. He suggests 
that L was put by Eratosthenes at 278 myriads of stades, not 78 ; 
but I am not clear that 78 is wrong. We have seen that, if we 
take mean figures, Aristarchus made the distance of the moon from 
the earth to be about 9^ times the earth's diameter. Now 252,000/77-, 
approximately 252,000/3^, is about 80,180, or roughly 8 myriads 
of stades; 9J times this is 76 myriads, and Eratosthenes' supposed 
figure of 780,000 is sufficiently close to this. According to 
Tannery's conjecture of 2,780,000 stades, the ratio L/2t would be 
nearly 34-7, which is greater than the values given to it by Hippar- 
chus, Posidonius, and Ptolemy, and also greater than the true value. 
With regard to Eratosthenes' estimate of S, Tannery points to 
Macrobius's statement that Eratosthenes said that 'the measure 
(mensura) of the earth multiplied 2η times will make the measure 
of the sun '. 4 The question here arises whether it is the solid 
contents of the two bodies or their diameters which are compared. 
Tannery takes the latter to be the meaning. If this is right, and 
if Eratosthenes took the value of J° for the apparent angular 
diameter of the sun discovered by Aristarchus, the circumference 
2ttS of the sun's orbit would be equal to 2η .2t . 720, which, if we 
put 3^ for π, would give 

S = 6185/ = 24,800 myriads of stades, nearly. 

1 Cf. Dreyer, Planetary Systems, p. 175. 

2 Aet. ii. 31. 3 (£>. G. 362-3). 

3 Tannery, 'Aristarque de Samos' in Mem. de la Soc. des sci.phys. et nat. de 
Bordeaux, 2 e ser., v, 1883, pp. 254, 255 ; Memoires scienttfiques, ed. Heiberg and 
Zeuthen, i, pp. 391-2. 

4 Macrobius, In somn. Scifi. i. 20. 9. 



ch.iv ARISTARCHUS'S CALCULATIONS 341 

But Hultsch Ί shows reason for believing that i mensura ' in the 
statement of Macrobius means solid content. One ground is the 
further statement of Macrobius that Posidonius's estimate of the size 
of the sun in terms of the earth was * many many times ' greater than 
that of Eratosthenes ( l multo multoque saepius ', sc. ' multiplicata '). 
But we shall find that Posidonius's figures lead to only about 39^ as 
the ratio of the diameter of the sun to that of the earth, which is 
not 'many many times' greater than 27. It seems therefore 
necessary to conclude, if Macrobius is to be trusted, that according 
to Eratosthenes s/t was equal to 3, not 2η. This would divide 
the value of £ by 9, and S/2t would be equal to 343^ instead of 

2>°9 2 h 

We are much better informed on the subject of Hipparchus's 

estimates of sizes and distances, thanks to the investigations of 
Hultsch, 2 who found in the commentaries of Pappus and Theon 
on chapter 1 1 of Book V of Ptolemy's Sy η taxis particulars as to 
which Ptolemy himself leaves us entirely in the dark. Ptolemy 
states that there are certain observed facts with regard to the 
sun and moon which make it possible, when the distance of one 
of them from the centre of the earth is known, to calculate the 
distance of the other, and that Hipparchus first found the dis- 
tance of the sun on certain assumptions as to the solar parallax, 
and then deduced the distance of the moon. According to the 
value assumed for the solar parallax (and Ptolemy says that there 
was doubt as to whether it was the smallest appreciable amount 
or actually negligible), Hipparchus deduced, of course, different 
figures for the distance of the moon. 3 Ptolemy does not state these 
figures, but Pappus supplies the deficiency. Pappus begins by 
saying that Hipparchus's calculation, depending mainly on the sun, 
was * not exact '. Next, he observes that, if the apparent diameter 
of the sun is taken to be very nearly the same as that of the moon 
at its greatest distance at the conjunctions, and if we are given the 
relative sizes of the sun and moon and the distance of one of them, 
the distance of the other is also given ; then, after paraphrasing 

1 Hultsch, Poseidonios icber die Grosse und Entfernung der Sonne, pp. 5, 6. 

2 Hultsch, ' Hipparchos iiber die Grosse und Entfernung der Sonne ' (Berichte 
der philologisch-historischen Classe der Kgl. Sachs. Gesellschaft der Wissen- 
schaften zu Leipzig, 7. Juli 1900). 

3 Ptolemy, Syntaxis, v. II, vol. i, p. 402, Heib. 



342 LATER IMPROVEMENTS ON part π 

Ptolemy's remarks above quoted, he proceeds as follows: ' In his first 
book about sizes and distances Hipparchus starts from this observa- 
tion : there was an eclipse of the sun which was exactly total in 
the region about the Hellespont, no portion of the sun being seen, 
whereas at Alexandria in Egypt about four-fifths only of its 
diameter was obscured. 1 From the facts thus observed he proves 
in his first book that, if the radius of the earth be the unit, the least 
distance of the moon contains 71, and the greatest 83 of these units ; 
the mean therefore contains ηη. After proving these propositions, 
he says at the end of the first book : " In this treatise I have carried 
the argument to this point. Do not, however, suppose that the 
theory of the distance of the moon has ever yet been worked out 
accurately in every respect ; for even in this question there is an 
investigation remaining to be carried out, in the course of which the 
distance of the moon will be proved to be less than the figure just 
calculated," so that he himself admits that he is not quite in a 
position to state the truth about the parallaxes. Then, again, he 
himself, in the second book about sizes and distances, proves from 
many considerations that, if we take the radius of the earth as the 
unit, the least distance of the moon contains 62 of these units, and its 
mean distance 6j±, while the distance of the sun contains 2,490. It 
is clear from the former figures that the greatest distance of the 
moon contains 72I of these units.' The figure of 2,490 for the 
distance of the sun has to be obtained by a correction of the Greek 
text. The later MSS. have ς or 90, but one MS. has Uq or 490. 
The 2,490 is credibly restored by Hultsch on the following grounds. 
Adrastus 2 and Chalcidius 3 tell us that Hipparchus made the sun 
nearly 1880 times the size of the earth, 4 and the earth about 2η times 
the size of the moon. The size means the solid content, and, the 
cube root of 1880 being approximately 12 J, we have approximately 

t\l\s = 1 : J : 12J 

1 This same eclipse is also mentioned by Cleomedes, De motu circulari, ii. 3, 
pp. 172. 22 and 178. 14, ed. Ziegler. 

2 Theon of Smyrna, p. 197. 8-12, ed. Hiller. 

3 Chalcidius, Ti?naeus, c. 91, p. 161. 

4 A less trustworthy authority, Cleomedes {Deniotu circulari, ii. 1, p. 152. 5-7), 
mentions a tradition that Hipparchus made the sun 1050 times as large as the 
earth. 



ch.iv ARISTARCHUS'S CALCULATIONS 343 

Now the mean distance of the moon is, according to Hipparchus, 
6j± times the earth's radius; assuming then that the apparent 
angular diameter of the sun and moon as seen from the earth is 
about the same, we find that 

S =67i*• 37 = 2491 J/. 
That is to say, S = 2490 /, nearly. It is clear, therefore, that β 
has fallen out of the text before υς, and the true number arrived at 
by Hipparchus was 2490. 

Thus Hipparchus made the distance of the moon equal, at the 
mean, to 33! times the diameter of the earth, and the distance of the 
sun equal to 1 ,245 times the diameter of the earth. As we said above, 
Ptolemy does not mention these figures of Hipparchus, much less 
does he make any use of them. Yet they are remarkable, because 
not only are they far nearer the truth than Aristarchus's estimates, 
but the figure of 1,245 f° r tne distance of the sun is much better 
than that of Ptolemy himself, namely 605 times the earth's 
diameter, or less than half the figure obtained by Hipparchus. 
Yet Hipparchus's estimate remained unknown, and Ptolemy's held 
the field for many centuries ; even Copernicus only made the 
distance of the sun to be equal to 750 times the earth's diameter, 
and it was not till 167 1-3 that a substantial improvement was made, 
observations of Mars carried out in those years by Richer enabling 
Cassini to conclude that the sun's parallax was about 9"«5, corre- 
sponding to a distance of the sun from the earth of 87,000,000 miles. 1 

Hultsch shows that the particular solar eclipse referred to by 
Hipparchus was probably that of 20th November in the year 
129 B.C., 2 and he concludes that the following year (128 B.C.) was 
the date of Hipparchus's treatise in two books l On the sizes and 
distances of the sun and moon '. 

Hipparchus, in his Geography, definitely accepted the estimate 
of 252,000 stades obtained by Eratosthenes for the circumference 
of the earth ; 3 if there is any foundation for the statement of Pliny 4 
that he added a little less than 26,000 stades to this estimate, making 
nearly 278,000, the explanation must apparently be that he stated 

1 Berry, A Short History of Astronomy •, 1898, pp. 205-7. 

2 Cf. Boll, Art. ' Finsternisse ' in Pauly-Wissowa's Real-Encyclopadie, vi. 2, 
1909, p. 2358. 

3 Strabo, ii. 5. 34, p. 132 Cas. 4 Pliny, Ν. H. ii. c. 108, § 247. 



344 LATER IMPROVEMENTS ON partii 

this number as a maximum, allowing for possible errors resulting 
from Eratosthenes' method; 1 but Berger considers that Pliny's 
statement is based on a misapprehension. 2 

Posidonius of Rhodes (135-51 B.C.) cannot be reckoned among 
astronomers in the strict sense of the term, but he dealt with astro- 
nomical questions in his work on Meteorology, and he wrote a 
separate tract on the size of the sun. 3 It was presumably in the 
latter work that he put forward a bold hypothesis as to the distance 
of the sun, which has the distinction of coming far nearer to the 
truth than the estimates of Hipparchus and all other ancient writers 
had done. 4 Cleomedes tells us that Posidonius supposed the circle 
in which the sun apparently moves round the earth to be 10,000 
times the size of a circular section of the earth through its centre. 
With this hypothesis he combined (says Cleomedes) the assumption 
which he took from Eratosthenes that at Syene (which is under 
the summer tropic) and throughout a circle round it with a diameter 
of 300 stades the upright gnomon throws no shadow (at noon). 
It follows from this that the diameter of the sun occupies a portion 
of the sun's circle 3,000,000 stades in length ; in other words, the 
diameter of the sun is 3,000,000 stades. 5 If we only knew the 
fraction of the circumference of the sun's circle occupied by the sun 
itself, we could calculate the circumference of the earth, and the 
absolute distance of the centre of the sun from the centre of the 
earth ; but Cleomedes gives us no information on this, and we have 
to go elsewhere for what we want — in this case to Pliny. Now Pliny 
says that according to Posidonius there is round the earth a height 
of not less than 40 stades, which is the region of winds and clouds, 
and beyond which there is pure air ; the distance from the belt of 
clouds, &c, to the moon is 2,000,000 stades, and the further distance 
from the moon to the sun is 500,000,000 stades. 6 This would give 

L — t— 2,000,040 stades, 

S—t = 502,000,040 stades. 

1 Tannery, Recherches sur Γ hist, de Vastronomie ancienne, p. 116. 

2 Berger, Gesch. der wissenschaftlichen Erdkunde der Griechen, pp. 413-14. 

3 Cleomedes, De motu circularly i. 11, p. 118. 4-6. 

4 On the whole of this subject, see Hultsch, ' Poseidonios iiber die Grosse 
und Entfernung der Sonne' (A ok. der Kgl. Gesellschaft der Wissenschaften zu 
Gottingen, Phil.-hist. Klasse, Neue Folge, Bd. I, Nr. 5), 1897. 

5 Cleomedes, ii. 1, p. 144. 22-146. 16; ibid. i. 10, pp. 96. 28-98. 5. 

6 Pliny, ii, c. 23, § 85. 



ch.iv ARISTARCHUS'S CALCULATIONS 345 

Dividing the latter figure by 10,000 we obtain, approximately, for 
the radius of the earth 

/ = 50,200 stades. 

Hultsch gives reason for thinking that the 500,000,000 stades 
should be the distance from the centre of the earth to the centre of 
the sun, not the distance from the moon to the sun ; the 40 stades 
representing the depth of the region of clouds, &c, is clearly 
negligible ; and, as Posidonius dealt in round figures, we may infer 
that his estimate of the earth's diameter would be 100,000 stades. 
If now we use the Archimedean approximation of 3^ for 77-, the 
circumference of the earth would on this basis be 314,285 stades; 
but we may, with some probability, suppose that Posidonius would 
take the round figure of 300,000 stades corresponding to π = 3, 
an approximation used by Cleomedes in another place. 1 

On the other hand, Cleomedes gives 240,000 stades as Posidonius's 
estimate of the earth's circumference based on the following assump- 
tions, (1) that the star Canopus, invisible in Greece, was just seen to 
graze the horizon at Rhodes as it rose and set again immediately, 
whereas its meridian altitude at Alexandria was 4 a fourth part of a 
sign, that is, one forty-eighth part of the zodiac circle ', (2) that the 
distance between the two places was considered to be 5,000 stades. 2 
The circumference of the earth was thus made out to be 48 times 
5,000 or 240,000 stades. But the estimate of the difference of lati- 
tude at i/48th of a great circle, or 7J , was very far from correct 
(the true difference of latitude is 5J only) ; indeed the effects of 
refraction at the horizon would inevitably vitiate the result of such 
an attempt at measurement of the angle in question as Posidonius 
was in a position to make. Moreover, the estimate of 5,000 stades 
for the distance was also incorrect; it was the maximum estimate 
put upon it by mariners, while some put it at 4,000 only, and 
Eratosthenes, by observations of the shadows cast by gnomons, 
found it to be 3,750 stades only. 3 The existence of the latter 
estimate of the distance between Rhodes and Alexandria seems to 
account for Strabo's statement that Posidonius favoured * the latest 
of the measurements which gave the smallest dimensions to the 

1 Cleomedes, De motu circularly i. 8, p. 78. 22-3. 

2 Ibid. i. 10, pp. 93. 26-94. 22. 

3 Strabo, ii. 5. 24, pp. 1 2 5-6 Cas. ; Berger, Gesck. derwissenschaftlichen Erdkunde 
der Griechen, p. 415. 



346 LATER IMPROVEMENTS ON part π 

earth \ namely about 180,000 stades ;* for 180,000 is 48 times 3,750, 
just as 240,000 is 48 times 5,000. Now Eratosthenes must presum- 
ably have arrived at his distance of 3,750 stades by means of 
a calculation based on his own estimate of the total circumference 
of the earth (250,000 or 252,000) and the observed angle represent- 
ing the difference of the inclination of the shadows thrown by the 
gnomon at the two places respectively. 2 We are not told what 
the angle was, but it can be inferred that it was 5! ° or 5 T V , because 

250,000 (252,000) : 3,750 = 360 : 5f° (5 T V). 

It is nothing short of extraordinary that Posidonius should have 
used the 3,750 stades without a thought of its origin and then 
calculated the circumference of the earth by combining the 3,750 
with an estimate of the corresponding angle which is so grossly 
erroneous (7J ). It may seem not less extraordinary that Ptolemy 
(following Marinus of Tyre) should have accepted without any 
argument or question Posidonius 's figure of 180,000 stades. But 
the explanation doubtless is that Ptolemy's stades were Royal 
stades of 210 metres (nearly ^th of a Roman mile) instead ot 
Eratosthenes' stades of 1574 metres ; for Ptolemy in his Geography 
says that the length of a degree is 500 stades, 3 whereas Eratosthenes 
made a degree contain about 700 stades. Thus, as Ptolemy's 
stades were to Eratosthenes' as 4 to 3, Ptolemy's estimate of the 
circumference of the earth would, in stades of Eratosthenes, be 
240,000, the same as the estimate attributed by Cleomedes to 
Posidonius. 

As we have seen, Pliny's account of Posidonius's estimates of the 
distances of the sun and moon leads to about 300,000 stades, and 
not 240,000, as the circumference of the earth. What is the 
explanation of the discrepancy ? Hultsch takes the 300,000 stades 
and the assumption that the sun's circle is 10,000 times as large as 
the circumference of the earth to be part of a calculation of the 
minimum distance of the sun, on the ground that Cleomedes goes 
on to say that ' it is indeed plausible that the sun's circle is not less 
than 10,000 times the circumference of the earth, seeing that the 
earth is to it in the relation of a point ; but it may also be greater 

1 Strabo, ii. 2. 2, p. 95 Cas. 2 Berger, op. cit., pp. 579, 580. 

3 Ptolemy, Geography, vii. 5. 12. 



ch.iv ARISTARCHUS'S CALCULATIONS 347 

still without our knowing it ', 1 But it is somewhat awkward to 
suppose with Hultsch that Posidonius is arguing, • 1 take the earth 
to be of the size attributed to it by Dicaearchus, namely 300,000 
stades in circumference, although this figure exceeds the truth ; 
but I am satisfied that, even if I take the circumference to he fully 
300,000 stades, I shall yet arrive at an estimate of the sun's distance 
which is less than the true distance.' The italics are mine, and 
represent the part of Hultsch's argument which seems to me 
doubtful. The use of an exaggerated estimate of the earth's 
circumference with a view to a minimum estimate of the sun's 
distance is so strange that I prefer to suppose that, in the develop- 
ment of the hypothesis about the sun's distance, Posidonius simply 
used Dicaearchus's figure for the earth's circumference without any 
arriere-pensee at all. 

In considering the origin of the bold hypothesis of Posidonius 
with regard to the sun's distance, it is necessary to refer to the 
hypotheses of Archimedes with regard to the size of the universe, on 
which in his Sand-reckoner he bases his argument that it is possible 
to formulate a system for expressing numbers as large as we please, 
say a number such as the number of the grains of sand which would 
be required to fill an empty space as large as our ' universe '. For 
the purpose which he has in view, Archimedes has of course to 
take what he considers to be outside or maximum measurements. 
Thus, whereas his predecessors had tried to prove the perimeter 
of the earth to be 300,000 stades, he will allow it to be as much 
as ten times that 4 and not greater', viz. 3,000,000 stades. Next, 
whereas Aristarchus had made the sun between 18 and 20 times as 
large as the moon, he will take it to be 30 times, but not greater, so 
that (the earth being greater than the moon) the sun will be less 
than 30 times the size of the earth. Archimedes proceeds to con- 
sider the size of the so-called ' universe ' and of the sun. He has 
explained that the ■ universe ' as commonly understood by astrono- 
mers is the sphere which has for its centre the centre of the earth 
and for its radius the distance between the centre of the earth and 
the centre of the sun, but that the sphere of the fixed stars is much 
greater than this so-called ' universe '. Considering now the sun 

1 Cleomedes, ii. 1, p. 146. 12-16. The text has μείζονα αυτόν οντά ή πάλιν 
/xei'oi/a, ' it may be greater, or again it may be less ' ; Hultsch rejects fj πάλιν 
μύονα as a gloss inconsistent with the trend of Cleomedes' argument. 



348 LATER IMPROVEMENTS ON part Π 

in relation to its orbit, a great circle of the so-called ' universe ', 
Archimedes found by a rough experiment (in confirmation of 
Aristarchus's discovery that the apparent angular diameter of the 
sun is Y -§ oth of four right angles) that the angle subtended by 
the sun's diameter is between x^th and ^o tn P art oi a right angle, 
or between 6-§6 tri an< ^ sio tn P art °f f° ur right angles. By means 
of this result he proves that the diameter of the sun is greater than 
the side of a chiliagon (or a regular polygon with 1,000 sides) 
inscribed in its orbit. The proof of this is very interesting because 
we there see Archimedes abandoning the traditional view that 
the earth is a point in relation to the sphere in which the sun 
moves 1 (Aristarchus assumed it to be so in relation even to the 
moon's sphere), and recognizing parallax in the case of the sun, 
apparently for the first time ; for, from the fact that the apparent 
diameter of the sun, as seen at its rising by an observer on the 
surface of the earth, subtends an angle less than e^gth and greater 
than g Jo tn °f f° ur right angles, he proves geometrically that the 
arc of the sun's orbit subtended by a chord equal to the diameter of 
the sun subtends at the centre of the earth an angle greater than 
¥ f 2 tn an d a fortiori greater than x^Vo tn °f f° ur right angles. 
Now, says Archimedes, since 

(perimeter of chiliagon inscribed in sun's orbit) 

< ι ,000 (diam. of sun) 

< 30,000 (diam. of earth), 

while the perimeter of any regular polygon of more than six sides 
is greater than 3 times the diameter of the circle in which it 
is described, it follows that 

(diameter of sun's orbit) < 10,000 (diam. of earth). 

Posidonius assumed, not that the diameter of the sun's orbit was 
less than 10,000 times the diameter of the earth, but that it was 
equal to (or not less than) 10,000 times the earth's diameter. But 
the origin of his ratio of 10,000 : 1 is sufficiently clear ; he took it 
from Archimedes. Similarly, the combination of the estimate of 
300,000 stades for the circumference of the earth with Erato- 
sthenes' assumption that the shadowless circle at Syene was 300 

1 Cf. Cleomedes, De 7iiotu circularly i. 11, pp. 108-12, ed. Ziegler. 



ch.iv ARISTARCHUS'S CALCULATIONS 349 

stades in diameter suggests that Posidonius likewise adopted from 
Archimedes the tqV o tn P art °f f° ur right angles as the apparent 
angular diameter of the sun, being satisfied to take Archimedes' 
minimum estimate as the actual figure. 

It remains to express Posidonius 's estimates of the sun's and 
moon's sizes and distances in terms of the earth's diameter. On 
the basis of his estimate of 240,000 stades for the circumference ot 
the earth, the earth's diameter, which we will call D, is 240,000/^ 
stades, or about 76,400 stades. 

Distance of sun = 500,000,000 2^/76,400 = about 6,545 D. 
Diameter of sun = 3,000,000 Z5/76400 = 39J D. 
Distance of moon = 2,000,000 D/76,400 = 26\ D. 
Diameter of moon = τ ^ (diameter of sun) = 0157 Z>, nearly. 

As Ptolemy gives none of the estimates which Pappus's com- 
mentary on the Syntaxis quotes from Hipparchus's treatise on the 
sizes and distances of the sun and moon, it was not unnatural to 
suppose, as Wolf did, 1 that the elaborate calculations in Ptolemy 
(v. 13-16) were referable to Hipparchus. This cannot be so as 
regards the results, as Hultsch has shown by means of Pappus's 
commentary, though doubtless Ptolemy may have been at least 
partially indebted to Hipparchus for the methods which he fol- 
lowed. The following are Ptolemy's results : 

The mean distance of the moon = 59 times the earth's radius. 2 

„ ,, ,, ,, sun = 1,210 ,, ,, ,, 

The diameter of the earth = 3§ times the diameter of the moon. 3 

i> i> )> sun = log- ,, „ ,, ,, ,, 

It follows that 

the diam. of the sun = about 5 \ times the diam. of the earth. 

I will conclude with Hultsch's final comparative table 4 of sizes 

1 Wolf, Geschichte der Astronomie, pp. 174 sqq. 

2 Ptolemy, Syntaxis, v. 15, p. 425. 17-20, Heib. 

3 Ibid., v. 16, p. 426. 12-15, Heib. 

* Hultsch, Hipparchos iiber die Grosse und Entfernung der Sonne, p. 199. 



35ο 



COMPARISON OF CALCULATIONS 



and distances in terms of the earth's mean diameter (=1,716 
geographical miles): 







Mean dis- 




Mean dis- 








tance of 


Diameter 


tance of 


Diameter 






moon from 


of moon 


sun from 


of sun 






earth 




earth 




According 


to Aristarchus 


91 


A - 0-36 


180 


6f 


») j> 


Hipparchus 


33l 


i - 033 


1245 


12 1 


j> j» 


Posidonius 


26£ 


A = 0*157 


6545 


39t 


j> » 


Ptolemy 


29I 


A = °*29 


605 


5Ϊ 


In reality 




30-2 


0*27 


1 1 726 


108-9 



ARISTARCHUS OF SAMOS 

ON THE SIZES AND DISTANCES OF 

THE SUN AND MOON 



TEXT, TRANSLATION, AND NOTES 



ΑΡΙΣΤΑΡΧΟΥ ΠΕΡΙ ΜΕΓΕΘΩΝ ΚΑΙ 
ΑΠΟΣΤΗΜΑΤΩΝ ΗΛΙΟΥ ΚΑΙ ΣΕΛΗΝΗΣ 

(ΥΠΟΘΕ2ΕΙ2) 

α . Την σελήνην πάρα του ηλίου το φω? λαμβάνειν. 
5 β'. Την γήν σημείου Τ€ και κέντρου λόγον εχειν προ? την 
της σελήνης σφαΐραν. 

γ '. "Οταν η σελήνη διχοτόμο? ήμΐν φαίνηται, νεύειν ει? 
την ήμετεραν όψιν τον διορίζοντα τό τε σκιερον και το 
λαμπρον τή? σελήνη? μεγιστον κύκλον. 

ίο tf. 'Οταν ή σελήνη διχοτόμο? ήμΐν φαίνηται, τότε αυτήν 
άπεχειν του ηλίου έλασσον τεταρτημορίου τω του τεταρ- 
τημορίου τριακοστω. 

ε'. Τό τή? σκιά? πλάτο? σεληνων είναι δύο. 
q '. Ύήν σελήνην ύποτείνειν ύπο πεντεκαιδεκατον μερο? 
ΐ5 ζωδίου. 

Έπιλογίζεται ουν το του ήλιου απόστημα άπο τή? γη? του τή? 
σελήνη? άποστήματο? μείζον μεν ή όκτωκαιδεκαπλάσιον, έλασσον 
δε ή είκοσαπλάσιον , δια τή? περί την διχοτομίαν ύποθεσεω?' τον 

[W = Wallis. F = Fortia d'Urban. Vat. = Cod. Vaticanus Graecus 204.] 

Ι. ΑΡΙΣΤΑΡΧΟΥ] ΑΡΙΣΤΑΡΧΟΥ ΣΑΜΙΟΥ W 3. (ΥΠΟΘΕΣΕΙΣ) addidi 

(cf. υποθέσεως 1. 1 8 infra ; νποτίθεται Pappus) : ΘΕΣΕΙΣ W 4. to] om. Pappus 
8. τε] om. Pappus 12. τριακοστω] τριακοοτημορίω Pappus l6. ουν] δη Pappus 
16, 17. το τον ήλίον . . . αποστήματος] τό τον ήλίον απόστημα του της σελήνης αποστή- 
ματος προς την γήν Pappus 1 8. είκοσαπλάσιον] είκοσιπλάσιον W δια της 
. . . νποθ^σεως] τούτο δε δια της περί την διχότομον νποθεσεως post 1. Ι, ρ. 354 
σελι)νης διάμετρον posuit Pappus 



ARISTARCHUS ON THE SIZES AND DISTANCES 
OF THE SUN AND MOON 

[hypotheses] 

ι . That the moon receives its light front the sun. 

2. That the earth is in the relation of a point and centre to the 
sphere in which the moon moves} 

3. That, when the moon appears to us halved, the great circle 
which divides the dark and the bright portions of the moon is 
in the direction of our eye? 

4. That, when the moon appears to us halved, its distance from 
the sun is then less than a quadrant by one-thirtieth of a 
quadrant? 

5. That the breadth of the {earth's) shadow is (that) of two 
moons. 

6. That the moon subtends one fifteenth part of a sign of the 
zodiac? 

We are now in a position to prove the following- propositions : — 

1. The distance of the sun from the earth is greater than 

eighteen times, but less than twenty times, the distance of the 

moon (from the earth) ; this follows from the hypothesis about 

the halved moon. 

1 Literally ( the sphere of the moon*. 

8 Literally 'verges towards our eye', the word veveiv meaning to 'verge' or 
1 incline \ What is meant is that the plane of the great circle in question passes 
through the observer's eye or, in other words, that his eye and the great circle 
are in one plane (cf. Aristarchus's own explanation in the enunciation of Prop. 5). 

s I.e. is less than 90 by i/3oth of 90 or 3 , and is therefore equal to 87 . 

4 I. e. i/i5th of 30 , or 2°. Archimedes in his Sand-reckoner (Archimedes, ed. 
Heiberg, ii, p. 248, 19) says that Aristarchus 'discovered that the sun appeared 
to be about 1 /720th part of the circle of the zodiac'; that is, Aristarchus dis- 
covered (evidently at a date later than that of our treatise) the much more 
correct value of \° for the angular diameter of the sun or moon (for he maintained 
that both had the same angular diameter : cf. Prop. 8). Archimedes himself 
in the same place describes a rough method of observation by which he inferred 
that the diameter of the sun was less than 1 /164th part, and greater than 
l/20oth part, of a right angle. Cf. pp. 311-12 ante. 

1410 A a 



354 



ON THE SIZES AND DISTANCES 



αύτον δε λόγον εχειν την του ήλιου διάμετρον προ? την τη? σελήνης 
διάμετρον την δε του ηλίου διάμετρον προ? την τή? γη? διάμετρον 
μείζονα μεν λόγον εχειν ή ον τα ιθ προ? γ, ελάσσονα δε ή ον μγ 
προ? q, δίά τον εύρεθεντο? περί τα αποστήματα λόγου, τη? (τε) 
5 περί την σκιάν ύποθεσεω?, και του την σελήνην ύπο πεντεκαιδεκατον 
μερο? ζωδίου ύποτείνειν. 



α . 



Αύο σφαίρα? ϊσα? μεν 6 αύτο? κύλινδρο? περιλαμβάνει, 
άνισου? δε δ αύτο? κώνο? την κορυφήν έχων προ? ttj 
ίο ίλάσσονι σφαίρα' και ή δια των κέντρων αυτών αγομένη 
εύθεΐα ορθή εστίν προ? εκάτερον τών κύκλων, καθ* ων 
εφάπτεται ή του κυλίνδρου ή ή του κώνου επιφάνεια τών 
σφαιρών. 

"Έ,στωσαν ϊσαι σφαΐραι, ων κέντρα έστω τα Α, Β σημεία, και 

1 5 επιζευχθεΐσα ή ΑΒ εκβεβλήσθω, και εκβεβλήσθω δια του ΑΒ 

επίπεδον ποιήσει δη τομά? εν ταΐ? σφαίραι? μεγίστου? κύκλου?. 





C 


Ύ 






Γ? 




ν\ κ 


Α 


α 


V 


Θ[ 


Β 


β 


Λ 


Μ 






Ιδ 


ν\ 









Ε* 



Η0 



Fig. 1 6. 

ποιείτω ουν του? ΤΔΕ, ΖΗΘ κύκλου?, καϊ ήχθωσαν άπο τών Α, Β 
ttj ΑΒ προ? ορθά? αϊ ΓΑΕ, ΖΒΘ, και επεζεύχθω ή ΓΖ. και επει 



I. €%€Lv την] εχ€ΐ και η Pappus διάμετρον] διάμ€τρος Pappus 3• μ*ίζ ονα 

μεν λόγον %χαν] iv μαίζονι λόγω Pappus τα] OlTl. Pappus ελάσσονα de] *ν 

(λάσσονι oi λόγω Pappus μγ] τά μγ Pappus 4• T *i s l Te )] ( rf ) addidi : καϊ τη* 
Pappus 6. inoreivetv ante νπο posuit Pappus 16. δη] δέ W 



OF THE SUN AND MOON 355 

2. The diameter of the sun has the same ratio (as aforesaid) 
to the diameter of the moon} 

3. The diameter of the sun has to the diameter of the earth 
a ratio greater than that which 19 has to 3, but less than that 
which 43 has to 6 ; this follows from the ratio thus discovered 
between the distances, the hypothesis about the shadow, and the 
hypothesis that the moon subtends one fifteenth part of a sign of 
the zodiac. 

Proposition i. 

Two equal spheres are comprehended by one and the same 
cylinder, and two unequal spheres by one and the same cone which 
has its vertex in the direction of the lesser sphere ; and the 
straight line drawn through the centres of the spheres is at right 
angles to each of the circles in which the surface of the cylinder, 
or of the cone, touches the spheres. 

Let there be equal spheres, and let the points A, Β bo. their 
centres. 

Let A Β be joined and produced ; 
let a plane be carried through A Β ; this plane will cut the spheres 
in great circles. 2 

Let the great circles be CDE, FGH. 

Let CAE, FBH be drawn from A, Β at right angles to A Β ; 
and let CF be joined. 

1 Pappus gives this second result immediately after the first result, i. e. before 
the parenthesis ' this follows from the hypothesis . . Λ This arrangement seems 
at first sight more appropriate, and Nizze alters his text accordingly. But 
I think it better to follow the above order which is that of the MSS. and Wallis. 
One consideration which weighs with me is that the second result does not 
follow from the hypothesis of the halved moon alone ; it depends on another 
assumption also, namely, that the sun and the moon have the same apparent 
angular diameter (see Prop. 8). 

a Literally ' it will make, as sections in the spheres, great circles \ and 
then, in the next sentence, Met it then make the circles CDE, FGH.' In 
translating these characteristic phrases, which occur very frequently, I wish 
I could have reproduced the Greek exactly, keeping the word ' sections ', but it 
becomes impossible to do so when the phrase is extended so as to distinguish 
several sections made by one plane, e.g. one section in one sphere, one section 
in another sphere, and one section in a cone : Thus i let it make, as sections, in 
the spheres, the circles CDE, FGH, and, in the cone, the triangle CEK ' 
(Prop. 2) would be intolerable, with or without the multitude of commas, 
whereas clearness and conciseness is easily secured by saying ' let it cut the 
spheres in the circles CDE, FGH and the cone in the triangle CEK \ 

A a 2 



35t 



ON THE SIZES AND DISTANCES 



ai Τ A, ZB ϊσαι τε και παράλληλοι εισιν, καϊ αί ΤΖ, ΑΒ άρα ϊσαι 
τε καϊ παράλληλοι εισιν. παραλληλόγραμμοι άρα εστίν το ΓΖΑΒ, 
και αι προ? τοις Τ, Ζ γωνίαι όρθαι έσονται- ώστε η ΤΖ των Τ ΔΕ, 
ΖΗΘ κύκλων εφάπτεται, εάν δη μενούση? τή? ΑΒ το ΑΖ παραλ- 
5 ληλόγραμμον και τα ΚΤΔ, ΗΖΑ ημικύκλια περιενεχθεντα ει? το 
αύτο πάλιν αποκατάσταση όθεν ήρξατο φερεσθαι, τα, μεν ΚΤΔ, 
ΗΖΑ ημικύκλια ενεγθήσεται κατά, των σφαιρών, το δε ΑΖ παραλ- 
ληλόγραμμον γεννήσει κύλινδρον, ου βάσει? έσονται οι περί δια- 
μέτρου? τα? ΤΕ, ΖΘ κύκλοι, ορθοί οντε? προ? την ΑΒ, δια το εν 

ίο πάση μετακινήσει διαμενειν τα? ΤΕ, Θ Ζ ορθά? τη ΑΒ. και 
φανερον οτι ή επιφάνεια αύτοΰ εφάπτεται των σφαιρών, επειδή ή ΤΖ 
κατά πασαν μετακίνησιν εφάπτεται τών ΚΤΔ, ΗΖΑ ημικυκλίων. 

'Έ,στωσαν δη αι σφαΐραι πάλιν, ων κέντρα έστω τά Α, Β, άνισοι, 
και μείζων ή? κεντρον το Α* λέγω οτι τά? σφαίρα? 6 αύτο? κώνο? 

ι 5 περιλαμβάνει την κορυφήν έχων προ? Trj ελάσσονι σφαίρα. 

'Έ,πεζεύγθω ή ΑΒ, και εκβεβλήσθω διά τη? ΑΒ επίπεδον 
ποιήσει δη τομά? εν ταΐ? σφαίραι? κύκλου?, ποιείτω του? Τ ΔΕ, 
ΖΗΘ' μείζων άρα ό Τ ΔΕ κύκλο? του ΗΖΘ κύκλου' ώστε και ή 
εκ του κέντρου του ΓΔΕ κύκλου μείζων εστί τη? εκ του κέντρου 

so του ΖΗΘ κύκλου, δυνατόν δή εστί λαβείν τι σημεΐον, ω? το Κ, ΐν 
rji ω? ή εκ του κέντρου του ΓΔΕ κύκλου προ? την εκ του κέντρου 




Fig. 17. 
του ΖΗΘ κύκλου, οΰτω? ή ΑΚ προ? τήν KB. έστω ουν ειλημμενον 
το Κ σημεΐον, και ήχθω ή ΚΖ εφαπτομένη του ΖΗΘ κύκλου, και 
επεζεύχθω ή ΖΒ, και διά του Α τη Β Ζ παράλληλο? ήχθω ή AT, 

6. άποκατασταθη] άποκατασττ} W ΙΟ. ΘΖ] Z0W II. ςφάπτ€ταϊ\ έφάπτηται W 
Ι3• Β ad init. Vat. et codd. Paris. δή] de W 17. τομας] corr. W : τομήν 

Vat. 18. κύκλο:] om. W ΗΖΘ] ΖΗΘ W 20. τό Κ] τ6 K6 Vat. 



OF THE SUN AND MOON $$ 7 

Then, since CA, FB are equal and parallel, therefore CF, A Β 
are also equal and parallel. 

Therefore CFAB is a parallelogram, 
and the angles at C, F will be right ; 
so that CF touches the circles CDE, FGH. 

If now, A Β remaining fixed, the parallelogram AF and the 
semicircles KCD, GFL be carried round and again restored to the 
position from which they started, the semicircles KCD, GFL will 
move in coincidence with the spheres 1 ; and the parallelogram AF 
will generate a cylinder, the bases of which will be the circles about 
CE, FH as diameters and at right angles to AB, because, through- 
out the whole motion, CE, HF remain at right angles to AB. 

And it is manifest that the surface of the cylinder touches the 
spheres, 

since CF, throughout the whole motion, touches the semicircles 
KCD, GFL. 

Again, let the spheres be unequal, and let A , Β be their centres ; 
let that sphere be greater, the centre of which is A. 

I say that the spheres are comprehended by one and the same 
cone which has its vertex in the direction of the lesser sphere. 

Let A Β be joined, and let a plane be carried through A Β ; 
this plane will cut the spheres in circles. 

Let the circles be CDE, FGH; 
therefore the circle CDE is greater than the circle GFH; so that 
the radius of the circle CDE is also greater than the radius of the 
circle FGH. 

Now it is possible to take a point, as Κ (on A Β produced), such 
that, as the radius of the circle CDE is to the radius of the circle 
FGH, sois AK to KB. 

Let the point Κ be so taken, and let KFbe drawn touching the 
circle FGH; 
let FB be joined, and through A let AC be drawn parallel to BE; 

1 The force of κατά here is very difficult to render. The Greek phrase 
(νίχθήσβται κατά τών σφαιρών means ' will be carried, or move, in the spheres ', 
that is, the circumferences of the semicircles will pass neither over nor under the 
surfaces of the spheres, but in coincidence with them throughout, in other words, 
they will by their revolution describe (as we say) the actual surfaces of the 
spheres. 



35* 



ON THE SIZES AND DISTANCES 



και επεζεύγθω ή ΓΖ. και επεί εστίν, ω? ή ΑΚ προ? την KB, ή 
Α Δ προ? την ΒΝ, ίση δε ή μεν ΑΔ τη AT, f) δε ΒΝ τη ΒΖ, 
εστίν άρα, ω? ή ΑΚ προ? την KB, ή AT προ? την ΒΖ. καϊ εστίν 
παράλληλο? ή Α Γ τη ΒΖ' ευθεία άρα εστίν ή ΓΖΚ. και 'ίστιν 
5 ορθή ή ύπο των ΚΖΒ- ορθή άρα και ή υπ ο των ΚΓΑ. εφάπτεται 
άρα η ΚΓ τον Τ ΔΕ κύκλου, ήγθωσαν δη αι ΓΑ, Ζ Μ επί την 
ΑΒ κάθετοι. εάν δη μενούση? τη? Κα τά τε ΖΓΔ, ΗΖΝ 
ημικύκλια και τα, ΚΓΑ, ΚΖΜ τρίγωνα περιενεγθεντα ει? το αύτο 
πάλιν άποκατασταθη όθεν ήρξατο φερεσθαι, τα μεν άΓΔ, ΗΖΝ 

ίο ημικύκλια ενεγβήσεται κατά των σφαιρών, το δε ΚΓΑ τρίγωνον και 
το ΚΖΜ γεννήσει κώνου?, ων βάσει? είσΐν οι περί διαμέτρου? τα? 
ΓΕ, Ζ θ κύκλοι, ορθοί οντε? προ? τον ΚΑ άξονα' κέντρα δε αυτών 
τα Α, Μ* και 6 κώνο? τών σφαιρών εφάψεται κατά την επιφάνειαν, 
επειδή και ή ΚΖΓ εφάπτεται τών ΈΓΔ, ΗΖΝ ημικυκλίων κατά 

ΐ5 πάσαν μετακίνησιν. 

β'. 

'Καν σφαίρα ύπο μείζονο? εαυτή? σφαίρα? φωτίζηται, 
μείζον ημισφαιρίου φωτισθήσεται. 

^,φαΐρα γάρ, η? κεντρον το Β, ύπο μείζονο? εαυτή? σφαίρα? 
ao φωτιζεσθω, η? κεντρον το Α' λίγω δτι το φωτιζομενον μερο? τη? 
σφαίρα?, η? κεντρον το Β, μεΐζόν εστίν ημισφαιρίου. 




Fig. ιδ. 
Ε7Γ6ί yap δύο άνισου? σφαίρα? ό αύτο? κώνο? περιλαμβάνει την 

κορυφή ν εγων προ? τη έλασσον ι σφαίρα, έστω 6 περιλαμβάνων τά? 

σφαίρα? κώνο?, και εκβεβλήσθω διά του άζονο? επίπεδον ποιήσει 

3 5 δή τομά? εν μεν ταΐ? σφαίραι? κύκλου?, εν δε τω κώνω τρίγωνον. 

8. ΚΓΛ] ΚΓΔ Vat. 9• άποκατασταθβ] άποκαταστβ W Ι φ ΒΓΔ] ΖΓΔ Vat. 
ΙΟ. β'] Γ Vat. 17. φωτίζηται] φωτίζεται W 22. κωρος] κόνος Vat. 



OF THE SUN AND MOON 359 

let CF be joined. 

Then since, as A Κ is to KB, so is AD to BN, 
while AD is equal to ^4 C, and 2W to BF, 
therefore, as A Κ is to .Ώ?, so is A C to i?/ 7 !. 

And A C is parallel to 2?jF; 
therefore CF.AT is a straight line. 

Now the angle KFB is right ; 
therefore the angle KCA is also right : 
therefore KC touches the circle CDF. 

Let CD, FMbe. drawn perpendicular to AB. 

If now, KO remaining fixed, the semicircles Ο CD, GFN and the 
triangles KCL, KFM be carried round and again restored to the 
position from which they started, the semicircles OCD, GFN will 
move in coincidence with the spheres ; and the triangles KCL and 
KFM will generate cones, the bases of which are the circles about 
CF, FH as diameters and at right angles to the axis KL, the 
centres of the circles being L, M. 

And the cone will touch the spheres along their surface, since 
KFC also touches the semicircles Ο CD, GFN throughout the 
whole motion. 



Proposition 2. 

If a sphere be illuminated by a sphere greater than itself, 
the illuminated portion of the former sphere will be greater than 
a hemisphere. 

For let a sphere the centre of which is Β be illuminated by 
a sphere greater than itself the centre of which is A, 

I say that the illuminated portion of the sphere the centre ot 
which is Β is greater than a hemisphere. 

For, since two unequal spheres are comprehended by one and 
the same cone which has its vertex in the direction of the lesser 
sphere, [Prop. 1] 

let the cone comprehending the spheres be (drawn), and let a plane 
be carried through the axis ; 
this plane will cut the spheres in circles and the cone in a triangle. 



3 6 ° 



ON THE SIZES AND DISTANCES 



ποιείτω ovv εν μεν ταΐς σφαίραις κύκλους τονς ΤΔΕ, ΖΗΘ, εν δε 
τω κώνω τρίγωνον το ΤΕΚ, φανερον δη οτι το κατά την ΖΗΘ 
περιφερειαν τμήμα της σφαίρα?, ου βάσις εστίν ό περί διάμετρον 
την Ζ Θ κύκλος, φωτιζόμενον μέρος εστίν ύπο τον τμήματος τον 

5 κατά την ΤΔΕ περιφερειαν, ου βάσις εστίν 6 wepl διάμετρον την 
ΓΕ κύκλο?, ορθός ων προς την ΑΒ ευθείαν καϊ γαρ ή ΖΗΘ 
περιφέρεια φωτίζεται ύπο τής ΤΔΕ περιφερείας' εσχαται γαρ 
ακτίνες είσιν αϊ ΤΖ, Ε Θ' και εστίν εν τω ΖΗΘ τμήματι το 
κεντρον τής σφαίρας το Β' ώστε το φωτιζόμενον μέρος τής σφαίρας 

ίο μείζον εστίν ήμισφαιρίον. 



'Έ.ν τή σελήντ] ελάχιστος κύκλος διορίζει το τε σκιερον 
και το λαμπρόν, όταν 6 περιλαμβάνων κώνος τον τε ήλιον 
και την σελήνην την κορνφήν εχη προς τή ημέτερα όψει, 

ΐ5 ν Εστω γαρ ή μεν ημέτερα 6ψις προς τω Α, ήλίον δε κεντρον το 
Β, σελήνης δε κεντρον, όταν μεν 6 περιλαμβάνων κώνος τον τε 
ήλιον και την σελήνην την κορνφήν εχτ) προς ττ) ημέτερα όψει, το Τ, 
όταν δε μή, το Δ' φανερον δη οτι τα. Α, Τ, Β επ* ευθείας εστίν, 
εκβεβλήσθω δια τής Α Β και τον Δ ση μείον επίπεδον' ποιήσει δή 

20 τομάς, εν μεν ταΐς σφαίραις κύκλονς, εν δε τοις κώνοις ευθείας, 
ποιείτω δε και εν τή σφαίρα, καθ' ής φέρεται το κεντρον τής σελήνης, 
κύκλον τον Τ Δ' το Α άρα κεντρον εστίν αύτον• τούτο γαρ υπόκειται' 
εν δε τω ήλίω τον ΕΖΡ κύκλον, εν δε τή σελήνη, όταν μεν 6 
περιλαμβάνων κώνος τον τε ήλιον και την σελήνην την κορνφήν εγτ\ 

25 ττρος τχι ημέτερα όψει, κύκλον τον ΚΘΑ, όταν δε μή, τον ΜΝΒ, 
εν δε τοις κώνοις ευθείας τα,ς ΕΑ, ΑΗ, ΠΟ, ΟΡ, άξονας δε τους 
ΑΒ, ΒΟ. και επεί εστίν, ως ή εκ τον κεντρον τον ΕΖΗ κύκλον 
προς την εκ τον κεντρον τον Θ ΚΑ, όντως ή εκ τον κεντρον τον ΕΖΗ 
κύκλον προς την εκ τον κέντρου τον ΜΝΒ' αλλ' ως ή εκ τον 



4• την ΖΘ] ZOW II. </] Δ Vat. 15. η^ου Sc] ηλίου W 

ι6. μ&\ om. W 21. 5e] δη W 25. ΚΘΑ] ΘΚΛ W 26. row] om. W 

27. κύκλου] om. W 



OF THE SUN AND MOON 361 

Let it cut the spheres in the circles CDE, FGH, and the cone in 
the triangle CEK, 

It is then manifest that the segment of the sphere towards the 
circumference FGH, the base of which is the circle about FH as 
diameter, is the portion illuminated by the segment towards the 
circumference CDE, the base of which is the circle about CE as 
diameter and at right angles to the straight line A Β ; 
for the circumference FGH is illuminated by the circumference 
CDE, since CF> EH are the extreme rays. 1 

And the centre Β of the sphere is within the segment FGH; 
so that the illuminated portion of the sphere is greater than a 
hemisphere. 

Proposition 3. 

The circle in the moon which divides the dark and the bright 
portions is least when the cone comprehending both the sun and 
the moon has its vertex at our eye. 

For let our eye be at A , and let Β be the centre of the sun ; 
let C be the centre of the moon when the cone comprehending both 
the sun and the moon has its vertex at our eye, and, when this is 
not the case, let D be the centre. 

It is then manifest that A, C, Β are in a straight line. 

Let a plane be carried through A Β and the point D\ this plane 
will cut the spheres in circles and the cones in straight lines. 

Let the plane also cut the sphere on which the centre of the 
moon moves in the circle CD ; 

therefore A is the centre of this circle, for this is our hypothesis 
[Hypothesis 2]. 

Let the plane cut the sun in the circle EFR y and the moon, when 
the cone comprehending both the sun and the moon has its vertex 
at our eye, in the circle KHL and, when this is not the case, in the 
circle Μ NO ; 

and let it cut the cones in the straight lines Ε A , A G, QP> PR, the 
axes being Α Β, Β P. 

Then since, as the radius of the circle EFG is to the radius ot 

the circle HKL, so is the radius of the circle EFG to the radius of 

the circle Μ NO, 

1 In Wallis's figure the letters F, Η are interchanged. With his lettering, the 
extreme rays should be CH, EF. I have given F, Η the positions necessary to 
suit the text, and my figure agrees with that of Vat. 



3 6 * 



ON THE SIZES AND DISTANCES 



κέντρου του ΕΖΗ κύκλου προς την έκ τον κέντρου τον ΘΑΚ κύκλου, 
ούτως ή ΒΑ προ? τηνΑΓ' ως δε ή έκ του κέντρου του ΕΖΗ κύκλου 
προς την έκ του κέντρου του ΜΝΈ κύκλου, ούτως έστιν ή ΒΟ προς 
την Ο Δ- και ως άρα ή ΒΑ προς την AT, ούτως ή ΒΟ προς την 
5 Ο Δ. και δκλόντι, ως ή Β Γ προς την ΓΑ, ούτως ή ΒΑ προς την 




Fig. 19. 

ΔΟ, και εναλλάξ, ως ή Β Γ προς την ΒΔ, ούτως ή ΓΑ προς την ΔΟ. 
καϊ 'έστιν έλάσσων η Β Γ της ΒΔ' κέντρον γαρ έστι το Α του Γ Δ 
κύκλου' έλάσσων άρα και ή ΑΓ της ΔΟ. καϊ 'έστιν ϊσος δ ΘΚΑ 
κύκλος τω ΜΝα κύκλω' έλάσσων άρα έστιν καϊ ή ΘΑ της Μα [, 

ίο δια το λήμμα]' ωστ€ και ό περί διάμςτρον την ΘΑ κύκλος 
γραφόμβνος, ορθός ων προς την ΑΒ, έλάσσων έστιν του περί διά- 
μ^τρον την ΜΗ κύκλου γραφομένου, ορθού προς την ΒΟ. αλλ* ό 
μ\ν πςρι διάμςτρον την ΘΑ κύκλος γραφόμςνος, ορθός ων προς την 
ΑΒ, ό διορίζων έστιν έν τ$ σελήνη το re σκιερον καϊ το λαμπρον, 

1 5 όταν ο περιλαμβάνων κώνος τον re ηλιον και την σελήνην την 

ι. τον ΕΖΗ] ΕΖΗ W rod ΘΛΚ] ΘΚΑ W $. διελόντι] διαψίθέντι 

W, qui lacunam post καί ope versionis Commandini expleverat 



OF THE SUN AND MOON 



363 



while, as the radius of the circle EFG is to the radius of the circle 

HLK, so is Β A to A C, 

and, as the radius of the circle EFG is to the radius of the circle 

Μ NO, so is Β Ρ to PD, 

therefore, as Β A is to A C, so is Β Ρ to PD, 

and, seftarando, as BC is to CA, so is BD to DP\ 

therefore also, alternately, as Β C is to BD, so is CA to DP, 

And BC is less than BD, for ^ is the centre of the circle CD ; 
therefore A C is also less than DP. 

And the circle HKL is equal to the circle Μ NO ; 
therefore ZfZ, is also less than MO [by the Lemma 1 ]. 

Accordingly the circle drawn about HL as diameter and at right 
angles to A Β is also less than the circle drawn about MO as 
diameter and at right angles to Β P. 

But the circle drawn about HL as diameter and at right angles 
to A Β is the circle which divides the dark and the bright portions 
in the moon when the cone comprehending both the sun and the 
moon has its vertex at our eye ; 



1 The promised Lemma (the equivalent of which is stated, rather than proved, 
in Euclid's Optics, 24) does not appear. Some of the MSS. have a scholium 
containing a rather clumsy proof. A shorter proof is that of Nizze. We can 
use one circle instead of two equal circles ; and we have to prove that, if A, Ρ 
are points on the radius produced, Ρ being further from the centre (C) than A 




Fig. 20. 



is, and if AH, AL be the pair of tangents from A, and PM, PO the pair of 
tangents from P, then MO > HL. 

By Eucl. vi. 8 and 17, CM 2 = CT. CP, and CH 2 = CS.CA; therefore 
CT. CP=CS.CA, or CA:CP = CT: CS. But CA < CP ; therefore CT < CS, 
so that the chord HL is less than the chord MO. 



3 6 4 



ON THE SIZES AND DISTANCES 



κορνφην εγτ) προ? Trj ημέτερα οψεΐ' ό δε περί διάμετρον την ME 
κύκλος, ορθό? ων προ? την ΒΟ, ό διορίζων εστίν kv τη σελήνη το τε 
σκιερον και το λαμπρόν, όταν δ περιλαμβάνων κώνο? τον τε ηλιον 
και την σελήνην μη εχη την κορυφην προ? τη ημέτερα όψει* ώστε 
5 ελάσσων κύκλο? διορίζει kv τβ σελήνη το τε σκιερον και το λαμπρόν, 
όταν ό περιλαμβάνων κώνο? τον τε ηλιον και την σελήνην την 
κορυφην εχη προ? Trj ημέτερα όψει. 

δ'. 
Ό διορίζων κύκλο? εν Trj σελήνη το τε σκιερον και το 
ίο λαμπρόν αδιάφορο? εστί τω εν τη σελήνη μεγίστω κύκλω 
προ? αϊσθησιν. 

ν Εστω yap η μεν ημέτερα οψι? προ? τω Α, σελήνη? δε κεντρον το 

Β, και επεζεύγθω η ΑΒ, και εκβεβλήσθω δια τη? ΑΒ επίπεδον 

ποιήσει δη τομήν εν Trj σφαίρα μεγιστον κύκλον. ποιείτω τον 

1 5 ΕΓΔΖ, εν δε τω κώνω ευθεία? τα? ΑΓ 9 ΑΔ, ΔΤ' ο άρα περί 

PC 





~Tfi 


D 


Αη___^ ^^\^ 


Β 






""""Ι® — — ^^ 


β 






\ν -~^^ 




y 

1 ι 


* 





Fig. 21. 
διάμετρον την ΤΔ, προ? ορθά? ων Trj ΑΒ, ό διορίζων εστίν εν Trj 
σελήνυ τ ° Τ€ σκιερον και το λαμπρόν. λέγω δη οτι αδιάφορο? εστί 
τω μεγίστω προ? την αϊσθησιν. 

ν Ηχ#ω yap δια τον Β Trj Τ Δ παράλληλο? ή ΕΖ, και κείσθω 

20 τή? ΔΖ ημίσεια εκατερα των ΗΚ, ΗΘ, και επεζεύγθωσαν αϊ KB, 

ΒΘ, ΚΑ, ΑΘ, ΒΔ. και επει υπόκειται ή σελήνη ύπο ιε μερο? 

Ι. την] τον Vat. 2. την] τον Vat. 3> 4• τ ° ν Τ€ Φ^ ιον καί Τ Ψ 0~έΚηνην\ om. W 

8• δ'] 6 Vat. 12. τω] το W 



OF THE SUN AND MOON 365 

and the circle about MO as diameter and at right angles to Β Ρ is 
the circle which divides the dark and the bright portions in the 
moon when the cone comprehending both the sun and the moon 
has not its vertex at our eye. 

Accordingly the circle which divides the dark and the bright 
portions in the moon is less when the cone comprehending both the 
sun and the moon has its vertex at our eye. 



Proposition 4. 

The circle which divides the dark and the bright portions in 
the moon is not perceptibly different from a great circle in the 
moon. 

For let our eye be at A, and let Β be the centre of the moon. 

Let A Β be joined, and let a plane be carried through A Β ; 
this plane will cut the sphere in a great circle. 

Let it cut the sphere in the circle ECDF and the cone in the 
straight lines AC, AD, DC 

Then the circle about CD as diameter and at right angles to A Β 
is the circle which divides the dark and the bright portions in the 
moon. 

I say that it is not perceptibly different from a great circle. 

For let EF be drawn through Β parallel to CD ; 
let GK, GHboUi be made (equal to) half of DF\ 
and let KB, BH, KA, AH, BD be joined. 

Then since, by hypothesis, the moon subtends a fifteenth part of 
a sign of the zodiac, 



3 66 



ON THE SIZES AND DISTANCES 



ζωδίου ύποτείνουσα, ή άρα ύπο ΓΑ Α γωνία βεβηκεν επι ιε μέρος 
ζω8ίον. το δε ιε' του ζωδίου του των ζωδίων όλου κύκλου εστίν ρπ , 
ώστε ή ύπο των ΓΑ Α γωνία βεβηκεν επι ρπ όλου του κύκλου' 
τεσσάρων άρα ορθών εστίν ή (ύπο) ΓΑ Α ρπ . δια δη τούτο ή ύπο 

δ ΓΑ Α γωνία εστίν με ορθής• καϊ εστίν αυτής ημίσεια ή ύπο ΒΑΑ 
γωνία' ή άρα ύπο των ΒΑΑ ημισείας ορθής εστί (με') μέρος, και 
επει ορθή εστίν ή ύπο των ΑΑΒ, ή άρα ύπο των ΒΑΑ γωνία προς 
ήμισυ ορθής μείζονα λόγον έχει ηπερ ή ΒΑ προς την Α Α, ώστε ή 
ΒΑ της ΔΑ ελάσσων εστίν ή με' μέρος, ώστε καϊ ή ΒΗ τής ΒΑ 

ίο πολλω ελάσσων εστίν ή με' μέρος. διελόντι ή Β Η τής Η Α 
ελάσσων εστίν η μδ μέρος, ώστε και η Β Θ τής ΑΘ πολλω 

6. ημισύας] corr. e μιας, ut videtur, Vat. et Paris. 2342 : μιας F Paris. 2366, 
2472 (?), 2488 Ο* 6 ') orn• Vat. et alii codd. μ*ρος] μς Paris. 2342 erasis 

litteris pos 10. SuXcWi] καϊ διαφ^θέντι W, qui lacunam post 10 η ope 

versionis Commandini expleverat 

1 This is a particular case of the more general proposition (similarly 
assumed by Archimedes in his Sand-reckoner) which amounts to the statement 
that, if each of the angles α, β is not greater than a right angle, and (X > /3, then 

tana a 
tan β > β ' 

The proposition is easily proved geometrically (cf. Commandinus on the 
passage of the Sand-reckoner). 

Let BC, Β A make with A CD the angles α, β respectively, and let BD be 
perpendicular to AD. 




Fig. 22. 

Now tan α = BD/CD, tan β = BD/AD. 

We have therefore to prove that 

AD : CD > (X : β. 



OF THE SUN AND MOON 367 

therefore the angle CAD stands on a fifteenth part of a sign. 

But a fifteenth part of a sign is 1/1 80th of the whole circle of the 
zodiac, 

so that the angle CAD stands on i/i8oth of the whole circle; 
therefore the angle CAD is i/i8oth of four right angles. 

It follows that the angle CAD is i/45th of a right angle. 

And the angle BAD is half of the angle CAD ; 
therefore the angle BAD is i/45th part of half a right angle. 

Now, since the angle ADB is right, 
the angle BAD has to half a right angle a ratio greater than that 
which BD has to DA. 1 

Accordingly BD is less than i/45th part of DA . 

Therefore BG is much less 2 than i/45th part of Β A, and, 
separando, BG is less than i/44th part of GA. 

Accordingly Β Η is also much less than 1/44Λ part of AH. 



Cut off AF equal to CD, and draw FE at right angles to AD and equal to 
BD. Join AE. 

Then AEAF= Z£CD = OC. 

Let EF meet AB in G. 

Since AE > AG > AF, the circle with A as centre and AG as radius will 
cut A Ε in Η and A F produced in K. 

Now I Ε AG : Ζ GAF = (sector HAG) : (sector GAK) 

< AEAG : A GAF 

< EG : GF. 
Componendo, Ζ EAF : Ζ GAF< EF : GF. 

But EF : GF= BD:GF=AD: AF= AD : CD. 

Therefore Oi : $<AD : CD, 

or AD'.CD>OL:$. 

In the particular application above made by Aristarchus OC = \ R, so that 
CD = BD. 

In this case therefore AD : DB > J R : Ζ BAD, 
or BD\DA<LBAD'.\R, 

that is to say, Ζ £,4 Ζ) :\R>BD: DA. 

2 ' Much less', ττολλώ έλάσσων = 'less by much', πόλλώ μύζων and ττολλώ 
(Κάσσων are the traditional expressions used by Euclid and Greek geometers in 
general for ( a fortiori greater' and '■ά. fortiori less '. In Euclid the expressions 
have generally been translated * much more then is . . . greater, or less, than '. 
But there is no double comparative in the Greek. The idea is that, if a is, let 
us say, a little greater than b, and if c is greater than a, then c must be much 
greater than b. 



3 68 



ON THE SIZES AND DISTANCES 



ελάσσων εστίν ή μδ' μύρο?, και έχει ή Β Θ προς την ΘΑ μείζονα 
λόγον ήπερ ή ύπο των ΒΑΘ προς την ύπο των ΛΒΘ• ή άρα ύπο 
των ΒΑΘ της νπο των ΑΒΘ ελάσσων εστίν ή μδ' μέρος, και εστίν 
της μεν νπο των ΒΑΘ διπλή ή ύπο των ΚΑΘ, της δε νπο των 
5 ΑΒΘ διπλή ή νπο των ΚΒΘ' ελάσσων άρα. εστίν και ή ύπο των 
ΚΑΘ τής ύπο των ΚΒΘ ή τεσσαρακοστοτεταρτον μέρος, άλλα, ή 
ύπο των ΚΒΘ ϊση εστίν ττ) ύπο των ΔΒΖ, τοντεστιν Trj ύπο των 
TAB, τοντεστιν Trj ύπο των ΒΑΔ' ή άρα ύπο των ΚΑΘ τής ύπο 
των ΒΑΔ ελάσσων εστ\ν ή μδ' μέρος, ή δε ύπο των ΒΑΔ (ημισείας) 
ίο ορθής εστίν (με') μέρος, ώστε ή ύπο των ΚΑΘ ορθής εστίν ελάσσων 

5, 6. ελάσσων . . . η~\ (ωστ€ η ΚΑΘ γωνία της ΚΒΘ γωνίας ελάσσων icrriv η) W 
9• {ήμισ(ίας), ΙΟ. (μβ'), supplevit W ΙΟ. (τοντίστι της ορθής <j' μέρος) post μέρος 

addidit W 






1 This is immediately deducible from a proposition given by Ptolemy 
{Syntaxis, I. io, pp. 43-4, ed. Heiberg). 

If two unequal chords are drawn in a circle, the greater has to the lesser 
a ratio less than the ciraimfercnce {standing) on the greater chord has to the 
circumference {standing) on the lesser. 

That is, if CB, Β A be unequal chords in a circle, and CB > Β A, then 
(chord CB) : (chord Β A) < (arc CB) : (arc Β A). 

Ptolemy's proof is as follows. 

Bisect the angle ABC by the straight line BD, meeting the circle again at D. 
Join AEC, AD, CD 




Then, since the angle ABC is bisected by BD t 

CD = AD. 
And CE>EA. 

Draw DF perpendicular to AEC. 



[Eucl., iii. 26, 29.] 
[Eucl., vi. 3.] 



OF THE SUN AND MOON 369 

And Β Η has to HA a ratio greater than that which the angle 
BAH has to the angle ABH} 

Therefore the angle BAH is less than i/44th part of the angle 
ABH 

And the angle KAH is double of the angle BAH, 
while the angle KB Η is double of the angle ABH; 
therefore the angle KAH is also less than i/44th part of the angle 
KBH 

But the angle KBH is equal to the angle DBF, that is, to the 
angle CDB, that is, to the angle BAD. 

Therefore the angle KAH is less than i/44th part of the angle 
BAD. 

But the angle BAD is i/45th part of half a right angle. 

Accordingly the angle Κ A His less than i/3Q6oth of a right angle. 2 

Now, since DA > DE > DF, the circle described with D as centre and DE 
as radius will cut AD between A and D, and will cut DF produced beyond F t 
Let the circle be drawn. 

Since the triangle A ED is greater than the sector DEG, and the triangle 
DEF is less than the sector DEH, 

Δ DEF : Δ DEA < (sector DEH) : (sector DEG). 
Therefore FE : Ε A < Δ FDE : Δ EDA . [Eucl., vi. 1 and 33.] 

Componendo, FA : Ε A < Δ FDA : Δ EDA. 

Doubling the antecedents, we have 

CA : AE < L CDA : ZADE, 
and, separando, CE: Ε A < I CD Ε : I EDA. 

But CE : Ε A = CB : Β A, 

and I CD Ε : Δ EDA = (arc CB) : (arc Β A). 

Therefore CB : Β A < (arc CB) : (arc Β A). 

[The proposition is easily seen to be equivalent to the statement that, if OL 
is an angle not greater than a right angle, and β another angle less than (X, then 

sin α (X -, 
sin β < /3' J 

Now, since ICDE = Δ CAB and ΔADE= ΔACB y in the same segments, 

we have 

CB :BA < A CAB : AACB, 

or, inversely, A Β : BC> IACB : ΔΒΑ C, 

which is the property assumed by Aristarchus. 



2 I'i'i- sAtj- 

1410 



Bb 



37© ON THE SIZES AND DISTANCES 

η γ~^£'. το δε νπο τηλικαύτης γωνία? δρώμενον μέγεθος άνεπαί- 
σθητδν εστίν τη ημετέρα όψει* και εστίν ίση ή ΚΘ περιφέρεια τη 
ΔΖ περιφέρεια' ετι άρα μάλλον ή ΔΖ περιφέρεια ανεπαίσθητος 
εστί τη ημετέρα, όψει. εαν γαρ επιζενχθη η ΑΖ, ή νπο των ΖΑΔ 
5 γωνία ελάσσων εστί της νπο των ΚΑΘ. το Δ άρα τω Ζ το αντο 
δό£ει είναι, διά τα αντα δη και το Τ τω Ε δόξει το αντο είναι• 
ώστε και ή Τ Δ τη Ε Ζ ανεπαίσθητος εστίν, και 6 διορίζων άρα εν 
τη σελήνη το τε σκιερον και το λαμπρον ανεπαίσθητος εστί τω 
μεγίστω. 

ΙΟ € . 

"Οταν η σελήνη διχοτόμος ήμΐν φαίνηται, τότε 6 

μέγιστος κύκλος δ πάρα τον διορίζοντα εν ttj σελήνη το 

τε σκιερον και το λαμπρον νεύει εις την ήμετέραν όψιν, 

τοντέστιν, 6 πάρα τον διορίζοντα μέγιστος κύκλος και ή 

ΐ5 ημετέρα οψις εν ενί είσιν έπιπέδω. 

Έ,πεϊ γαρ διχοτδμον ονσης της σελήνης φαίνεται 6 διορίζων το τε 

λαμπρον και το σκιερον της σελήνης κύκλος νεύων εις την ήμετέραν 

όψιν, και αντω αδιάφορος δ πάρα τον διορίζοντα μέγιστος κύκλος, 

όταν άρα ή σελήνη διχοτόμος ήμΐν φαίνηται, τότε δ μέγιστος κύκλος 

2ο ό πάρα, τον διορίζοντα νενει εις την ήμετέραν όψιν. 

ς • 
Ή σελήνη κατώτερον φέρεται τον ήλίον, και διχοτόμος 
ονσα έλασσον τέταρτη μορίον απέχει άπο τον ήλίον. 

"Εστω γαρ ή ημετέρα 8ψις προς τω Α, ήλίον δε κέντρον το Β, και 

3 5 έπιζενχθεϊσα ή ΑΒ έκβεβλήσθω, και έκβεβλήσθω δια, της ΑΒ και 

τον κέντρον της σελήνης διχοτόμον ούσης επίπεδον ποιήσει δή 

το μην εν τη σφαίρα, καθ* ή ς φέρεται το κέντρον τον ήλίον, κύκλον 

*• /Γ^£Ί /Γ'-ρ'£' Vat. : /y~^£' μέρος W η. ανεπαίσθητος] si C Vat. J,%. καϊ 

6 διορίζων αρα εν . . . ανεπαίσθητος εστί] (ό αρα διορίζων κύκλος εν . . . αδιάφορος 
€στι προς α'ίσθησιν) supplevit W, qui lacunam in suo cod ice animadverterat 

IO. e] f Vat. 13. λαμπρον] λαμπρον αυτού W : λαμπρον αυτής Nizze 

18. αδιάφορος] αδιάφορος εστίν W Ι9• φαίνηται] WF : φανήται Vat. 

21. $■'] om. Vat. 22. φέρεται] W F Paris. 2364, 2472 (?) : φαίνεται Vat. (in 
ras. sed ν quasi in ρ mutato) Paris. 2366. 24. τώ] τό W 



OF THE SUN AND MOON 371 

But a magnitude seen under such an angle is imperceptible to 
our eye. 

And the circumference KH is equal to the circumference DF ; 
therefore still more is the circumference DF imperceptible to 
our eye ; 
for, if AF be joined, the angle FAD is less than the angle Κ AH} 

Therefore D will seem to be the same with F. 

For the same reason, C will also seem to be the same with E. 

Accordingly CD is not perceptibly different 2 from EF. 

Therefore the circle which divides the dark and the bright por- 
tions in the moon is not perceptibly different from a great circle. 

Proposition 5. 

When the moon appears to us halved, the great circle parallel 
to the circle which divides the dark and the bright portions in 
the moon is then in the direction of our eye ; that is to say, the 
great circle parallel to the dividing circle and our eye are in one 
plane. 

For since, when the moon is halved, the circle which divides the 
bright and the dark portions of the moon is in the direction of our eye 
[Hypothesis 3], while the great circle parallel to the dividing circle 
is indistinguishable from it, 

therefore, when the moon appears to us halved, the great circle 
parallel to the dividing circle is then in the direction of our eye. 

Proposition 6. 

The moon moves {in an orbit) lower than (that of) the sun, and, 
when it is halved, is distant less than a quadrant from the sun. 

For let our eye be at A , and let Β be the centre of the sun ; let 
A Β be joined and produced, and let a plane be carried through 
A Β and the centre of the moon when halved ; 
this plane will cut in a great circle the sphere on which the centre 
of the sun moves. 

1 Pappus (pp. 560-8, ed Hultsch) gives an elaborate proof of this proposition 
depending on two lemmas ; the proof, however, in the text as we have it, contains 
a serious flaw (p. 568. 2-3). But the truth of the assumption in Aristarchus's 
particular case is so obvious as scarcely to require proof. 

2 ανεπαίσθητος is strangely used with dat. as if equivalent to ανεπαισθήτως 
Βιάφορος or αδιάφορος προς αΐσθησιν, ' imperceptibly different from '. 

Β b 2, 



I 



37* 



ON THE SIZES AND DISTANCES 



μεγιστον. ποιείτω ουν τον ΤΒΔ κύκλον, καϊ άπο του Α τη ΑΒ 
προς όρθάς ήχθω ή ΓΑΔ• τεταρτημορίου άρα εστίν ή ΒΔ περι- 
φέρεια, λέγω οτι ή σελήνη κατώτερον φέρεται του ηλίου, και 
διχοτόμος ούσα έλασσον τεταρτημορίου απέχει άπο του ήλιου, τουτ- 
5 εστίν, οτι το κεντρον εστίν αυτής μεταξύ των ΒΑ, Α Δ ευθειών 
και τής ΔΕ Β περιφερείας. 

Εί γαρ μή, έστω το κεντρον αυτής το Ζ μεταξύ των ΔΑ } Α Α 
ευθειών, και επεζεύχθω ή ΒΖ. ή Β Ζ άρα άξων εστίν του περι- 




Fig. 24. 

λαμβάνοντος κώνου τον τε ήλιον και την σελήνην, και γίνεται ή 
ίο ΒΖ ορθή προς τον διορίζοντα εν τη σελήνη τό τε σκιερον και το 
λαμπρον μεγιστον κύκλον. έστω ουν ο μέγιστος κύκλος εν τη 
σελήνη δ παρά τον διορίζοντα τό τε σκιερον και το λαμπρον 6 ΗΘΚ. 
καϊ επει διχοτόμου οϋσης τής σελήνης ό μέγιστος κύκλος ό παρά 
τον διορίζοντα εν τη σελήνη τό τε σκιερον και το λαμπρον και ή 
ΐ5 ημέτερα οψις εν ενί είσιν επίπεδο), επεζεύχθω ή ΑΖ* ή ΑΖ άρα εν 



Ι. τον] τό Vat. 



4. ΤΚασσον] Ζλαττον Vat. 



OF THE SUN AND MOON 373 

Let it cut it in the circle CBD ; and from A let CAD be drawn 
at right angles to AB. 

Then the circumference BD is that of a quadrant. 

I say that the moon moves (in an orbit) lower than (that of) the 
sun, and, when halved, is distant less than a quadrant from the sun ; 
that is to say, its centre is between the straight lines BA y AD and 
the circumference DEB. 

For, if not, let its centre F be between the straight lines DA , 
AL, and let BF be joined ; 

then BF is the axis of the cone which comprehends both the sun 
and the moon, 

and BF is at right angles to the great circle * which divides the 
dark and the bright portions in the moon. 

Let the great circle in the moon parallel to the circle which 
divides the dark and the bright portions be GHK\ 2 then since, 
when the moon is halved, the great circle parallel to the circle which 
divides the dark and the bright portions in the moon and our eye 
are in one plane [Prop. 5], let AF be joined. 

Therefore AFis in the plane of the circle KGH. 



1 It is of course not actually a great circle, but a circle parallel to a great 
circle, which is however so close to it as to be indistinguishable from a great 
circle so far as our vision of it is concerned [Prop. 4]. The expression is there- 
fore excusable, as in Hypothesis 3 ; there is no need to omit μ^ιστον from the 
text as Nizze does. 

2 I have drawn the circle GHK and the other circles representing the sections 
of the moon as they are drawn in Wallis's figures ; but I think the circles in the 
moon defining the dark and bright portions and, by hypothesis, in the same 
plane with our eye would be better represented by the dotted circles which 
I have added to the figure. 



374 ON THE SIZES AND DISTANCES 

τω του ΚΗΘ κύκλου εστίν επιπεδω. καϊ εστίν η Β Ζ τω ΚΘΗ 
κύκλω προς ορθά?, ώστε και τη ΑΖ• ορθή άρα εστίν ή ύπο ΒΖΑ 
γωνία, άλλα και αμβλεία ή ύπο των ΒΑΖ• όπερ αδύνατον, ουκ 
άρα το Ζ ση μείον kv τω ύπο την ΔΑΑ γωνίαν τόπω εστίν. 

5 Αεγω οτι ούδε επι της ΑΔ. ει γαρ δυνατόν, έστω το Μ, και 
πάλιν επεζεύχθω ή ΒΜ, και έστω μέγιστος κύκλος ο πάρα τον 
διορίζοντα, ου κεντρον το Μ. κατά τα αύτα δη δειχθήσεται η ύπο 
ΒΜΑ γωνία ορθή προς τον μεγιστον κύκλον άλλα και ή ύπο των 
ΒΑΜ' όπερ αδύνατον, ούκ άρα επι της ΑΔ το κεντρον εστί της 

ίο σελήνης διχοτόμου ούσης' μεταξύ άρα των ΑΒ, ΑΔ εστίν. 

Λέγω δη οτι και εντός της ΒΔ περιφερείας, ει γαρ δυνατόν, 
έστω έκτος κατά, το Ν, και τα αύτα κατεσκευάσθω. δειγθήσεται δη 
η ύπο των ΒΝΑ γωνία ορθή' μείζων άρα εστίν ή ΒΑ της ΑΝ. 
ίση δε ή ΒΑ τη ΑΕ' μείζων άρα εστίν και ή ΑΕ της ΑΝ• όπερ 

ΐ5 αδύνατον, ούκ άρα το κεντρον της σελήνης διχοτόμου ούσης έκτος 
εσται της ΒΕΔ περιφερείας, ομοίως δειγθήσεται οτι ούδε επ' 
αυτής της ΒΕΔ περιφερείας' εντός άρα. η άρα σελήνη κατώτερον 
φέρεται του ηλίου, και διχοτόμος ούσα έλασσον τεταρτημορίου 
άπεχει άπο του ηλίου. 



Ι. ΚΘΗ] ΚΗΘ W 1. νπο] νπο (των) W 4• την AAA γωνίαν] 

inusitato sane dicendi more : την (ύττό των) ΔΑΑ γωνίαν W, sed dubito an ipse 
Aristarchus νπο την νπο των scripserit II. ΒΑ] ΒΕΑ W 1 2. κατ€- 

σκενάσθω] κατασκευασθώ Vat. 1 3. γωνία] om. W 1 4. καϊ] om. W 

17. evrbs] W F Paris. 2364, 2472 (?) : έκτος Vat. in ras., Paris. 2363 



1 The phrase in the Greek text, κατά τα αντα. δή δειχθήσεται η νπο ΒΜΑ γωνία 
ορθή προς τον μίγιστον κύκλον, is strange. Literally this would appear to mean 
c In the same way it can be proved that the angle BMA is at right angles to 
the great circle ', but this is intolerable. If we took ' the angle BMA ' to be 
the plane of the angle, the expression would be possible, but it would not give 
the meaning which is required, namely that the angle BMA is a right angle 
because BM is at right angles to the plane of the circle and therefore to any 
straight line in the plane of the circle, such as AM, passing through M. The 



OF THE SUN AND MOON 375 

And BF is at right angles to the circle KHG, and therefore to 
AF; therefore the angle BFA is right. 

But the angle BAF is also obtuse : which is impossible. 

Therefore the point F is not in the space bounded by the angle 
DAL, 

I say that neither is it on AD. 

For, if possible, let it be M; and again let Β Μ be joined, and let 
the great circle parallel to the dividing circle be taken, its centre 
being M. 

Then, in the same way as before, it can be proved that the angle 
BMA [made with the great circle] λ is right. 

But the angle BAM is so also : which is impossible. 

Therefore the centre of the moon, when halved, is not on AD. 

Therefore it is between Λ Β and AD. 

Again, I say that it is also within the circumference BD. 

For, if possible, let it be outside, at N; 
and let the same construction be made. 

It can then be proved that the angle BNA is right ; therefore Β A 
is greater than AN. 

But Β A is equal X.o AE\ 
therefore A Ε is also greater than AN: which is impossible. 

Therefore the centre of the moon, when halved, will not be out- 
side the circumference BED. 

Similarly it can be proved that neither will it be on the circum- 
ference BED itself. 

Therefore it will be within. 

Therefore, &c. 



words προς τον μ^ιστον κυκλον are in fact not wanted, and, if they are 
retained, cannot be taken with ορθή in the sense of ' at right angles to the great 
circle' ; they can only be taken closely with γωνία and as meaning ' towards the 
great circle ', or ' made with the great circle '. But, as the words do not occur in 
the corresponding passage about the angle BNA lower down, I think they 
should be struck out, as an interpolation by some one who thought the inference 
wanted some further explanation but failed to supply it intelligibly. 



376 ON THE SIZES AND DISTANCES 

c 

To απόστημα ο απέχει ό ήλιο? άπο τή? γη? τον άπο- 
στήματο? ου απέχει ή σελήνη άπο τη? γη? μείζον μέν 
εστίν η όκτωκαιδεκαπλάσιον, 'έλασσον δε ή εικοσαπλάσιον. 

5 *Έστω γαρ ηλίου μεν κέντρον το Α, γη? δε το Β, καϊ επιζευχθεΐσα 
ή ΑΒ έκβεβλήσθω, σελήνη? δε κέντρον διχοτόμου οϋση? το Γ, και 
έκβεβλήσθω δια, τη? ΑΒ καϊ του Τ επίπεδον, και ποιείτω τομήν kv 
τη σφαίρα, κα& ή? φέρεται το κέντρον του ήλιου, μέγιστον κύκλον 
τον ΑΔΕ, και έπεζεύχθωσαν αϊ AT, ΓΒ, καϊ έκβεβλήσθω ή ΒΓ 

ίο επι το Α. εσται δή, δια το το Τ σημεΐον κέντρον είναι τή? σελήνη? 
διχοτόμου οϋση?, ορθή ή ύπο των ΑΤΒ. ηχθω δή άπο του Β τη 
ΒΑ προ? ορθά? ή BE. εσται δή ή ΕΔ περιφέρεια τή? ΕΔΑ 
περιφέρεια? λ ■ υπόκειται γαρ, όταν ή σελήνη διχοτόμο? ήμΐν φαί- 
νηται, άπέχειν άπο του ήλιου έλασσον τεταρτημορίου τω του 

ΐ5 τεταρτημορίου λ• ώστε και ή ύπο των ΕΒΤ γωνία ορθή? εστί λ. 
συμπεπληρώσθω δή το ΑΕ παραλληλόγραμμον, και επεζεύχθω ή 
ΒΖ. εσται δή ή ύπο των ΖΒΕ γωνία ημίσεια ορθή?, τετμήσθω 
ή ύπο των ΖΒΕ γωνία δίχα τή Β Η ευθεία' ή άρα ύπο των ΗΒΕ 
γωνία τέταρτον μέρο? εστίν ορθή?. άλλα, και ή ύπο των ΔΒΕ 

20 γωνία λ' εστί μέρο? ορθή?' λόγο? άρα τή? ύπο των ΗΒΕ γωνία? 
προ? τήν ύπο των ΔΒΕ γωνίαν (εστίν) ον (έχει) τα ιε προ? τά δύο' 
οΐων γάρ εστίν ορθή γωνία £, τοιούτων εστίν ή μεν ύπο των ΗΒΕ ιε, 
ή δε ύπο των ΔΒΕ δύο. και επει ή HE προ? τήν ΕΘ μείζονα 
λόγον έχει ήπερ ή ύπο των ΗΒΕ γωνία προ? τήν ύπο των ΔΒΕ 



3• ον] δ W F Nizze, sed nihil mutandum 4. ακοσαπλάσιον] €Ϊκοσιπ\άσιον 

W 6. τό Γ] <?στω> τό Γ Nizze 9. ΒΓ] ΓΒ W 12. BE] add. 

και (κβεβλήσθω η ΒΓ eVi το Δ Vat. Paris. 2364, 2366, 2472 (?) Ι3• λ Ί τ Ρ ια ~ 

κοστόν W Ι4• τω] om. W Ι5• λ'] τριακοστά) W των] την Vat. 

λ'] τριακοστόν W 1 8. νπο των (ad init.)] νπο W 20. λ'] τριακοστόν W 

21. γωνίαν] γωνίαν (έστιν) Nizze e^ei] om. Vat. 23. ή de νπο των] ή de W 



OF THE SUN AND MOON 377 

Proposition 7. 

The distance of the sun from the earth is greater than eighteen 
times, but less than twenty times, the distance of the moon from 
the earth. 

For let A be the centre of the sun, Β that of the earth. 

Let A Β be joined and produced. 

Let C be the centre of the moon when halved ; 
let a plane be carried through A Β and C, and let the section made 
by it in the sphere on which the centre of the sun moves be the 
great circle ADE. 

Let A C, CB be joined, and let BC be produced to D. 

Then, because the point C is the centre of the moon when 
halved, the angle A CB will be right. 

Let BE be drawn from Β at right angles to Β A ; 
then the circumference ED will be one-thirtieth of the circumterence 
EDA\ 

for, by hypothesis, when the moon appears to us halved, its dis- 
tance from the sun is less than a quadrant by one-thirtieth of a 
quadrant [Hypothesis 4]. 

Thus the angle EBC is also one- thirtieth of a right angle. 

Let the parallelogram A Ε be completed, and let BE be joined. 

Then the angle FBE will be half a right angle. 

Let the angle FBE be bisected by the straight line BG ; 
therefore the angle GBE is one fourth part of a right angle. 

But the angle DBE is also one thirtieth part of a right angle ; 
therefore the ratio of the angle GBE to the angle DBE is that 
which 1 5 has to 2 : 

for, if a right angle be regarded as divided into 60 equal parts, the 
angle GBE contains 15 of such parts, and the angle DBE 
contains 2. 

Now, since GE has to EH a ratio greater than that which the 
angle GBE has to the angle DBE, 1 

1 The proposition assumed is again the equivalent of the fact that > — > 

where each of the angles OL, β is not greater than a right angle and Οι > β. (Cf. 
note on pp. 366-7, above.) Let the angles α, β be the angles GBE, HBE 
respectively in the subjoined figure (Fig. 26). Let GE be perpendicular to BE 



378 



ON THE SIZES AND DISTANCES 



γωνίαν, ή άρα HE προ? την Ε Θ μείζονα λόγον έχβί ηπ€ρ τά ΐ€ 
προ? τα, β. καϊ 4πα ίση ίστιν ή BE τβ ΕΖ, καϊ ίστιν ορθή η 
προ? τω Ε, το άρα άπο τη? Ζ Β του άπο BE διπλάσιόν έστιν 
ω? δ\ το άπο ΖΒ προ? το άπο ΒΕ } οντω? ίστι το άπο Ζ Η προ? το 
5 άπο ΗΕ• το άρα άπο ΖΗ τον άπο HE διπλάσιόν έστι. τα δ\ μθ 
των Κ€ ίλάσσονά ίστιν ή διπλάσια, ωστ€ το άπο ΖΗ προ? το άπο 




Fig. 25. 

HE μβίζονα λόγον 'έχει ή (ον τα) μθ προ? Κ€' και ή ΖΗ 

άρα προ? την HE μείζονα λόγον %yei η (ον) τα £ προ? τα €• και 

σννθέντι ή ΖΕ άρα προ? την ΕΗ μείζονα λόγον '4χ€ΐ η ον τα ιβ 

ίο προ? τά €, τοντίστιν, ή ον (τα) λς* προ? τα ie. ίδ^ίχθη <5e και ή 

2. β] δύο W 3• BE] της BE W 7• (ον ™>] om • Vat. κ*] τα 

wW ' 8. (bv)] om. Vat. 9• *Χ«] έχουσα W ΙΟ. (τά)] 

om. Vat. δε] δ;) W 



OF THE SUN AND MOON 379 

therefore GE has to EH a. ratio greater than that which 15 has to 2. 

Next, since BE is equal to EF, and the angle at Ε is right, 
therefore the square on FB is double of the square on BE, 

But, as the square on FB is to the square on BE, so is the 
square on FG to the square on GE ; 
therefore the square on FG is double of the square on GE. 

Now 49 is less than double * of 25, 
so that the square on FG has to the square on GE a ratio greater 
than that which 49 has to 25 ; 

therefore FG also has to GE a ratio greater than that which 7 
has to 5. 

Therefore, componendo, FE has to EG a ratio greater than that 
which 12 has to 5, that is, than that which 36 has to 15. 

and let it meet BH in H. Let a circle be described with Β as centre and Β Η 
as radius, meeting BG in Ρ and BE produced in Q. 




EQ 



Fig. 26. 



Then Δ GBH : Δ HBE > (sector PBH) : (sector HBQ) ; 

therefore GH : HE > Ζ GBH : Ζ HBE, 

and, componendo, GE : //£ > Ζ GBE : Ζ HBE. 

1 Aristarchus here uses the well-known Pythagorean approximation to V2 , 
namely \ , one of the first of the successive approximations obtained by the 
development of the system of ' side- ' and ' diagonal- ' numbers (as to which 
see Theon of Smyrna, pp. 43, 44, ed. Hiller, and Proclus, Com?n. in Platonis rem- 
publicum, ed. Kroll, vol. ii, pp. 24, 25, 27-9, 393-400). The approximation 
\ is alluded to by Plato in the Republic, 546 c. Plato there speaks of the 
diagonal of the square, the side of which contains 5 units, and contrasts the 
' irrational diameter of 5 ' (άρρητος διάμετρος της πεμπάδος), which is of course 
>v/(5o),with the 'rational diameter' {ρητή διά /zerpos), which is the square root of 
50 less a single unit, i.e. the square root of 49. 



380 ON THE SIZES AND DISTANCES 

HE προ? την Ε Θ μείζονα λόγον άγουσα ή ον τα ιε προς τα δνο' 
δι ίσον άρα ή Ζ Ε προ? την Ε Θ μείζονα λόγον έχει ή ον τα λ<? 
προ? τα δυο, τουτέστιν, ή ον τα, ιη προ? α• η άρα ΖΕ τη? Ε Θ 
μείζων εστίν ή ιη. ή δε ΖΕ ϊση εστίν τη BE' και ή BE άρα τη? 
5 ΕΘ μείζων εστίν η ιη• πολλω άρα ή Β Θ τη? ΘΕ μείζων εστίν η 
ιη. αλλ* ώ? ή ΒΘ προ? την ΘΕ, οντω? εστίν ή ΑΒ προ? την ΒΤ, 
δια την ομοιότητα των τριγώνων και η ΑΒ άρα τη? ΒΤ μείζων 
εστίν η ιη. και εστίν ή μεν ΑΒ το απόστημα δ άπεχει ό ήλιο? άπο 
τη? γη?, ή δε ΤΒ το απόστημα ο απέχει ή σελήνη άπο τη? γη?' το 

ίο άρα απόστημα ο απέχει ό ήλιο? άπο τή? γη? τον άποστήματο?, ον 
απέχει ή σελήνη άπο τή? γη?, μείζον εστίν ή ιη. 

Λέγω δη οτι και έλασσον ή κ. ήχθω γαρ δια τον Δ τή ΕΒ 
παράλληλο? ή ΔΚ, και περί το Δ KB τρίγωνον κύκλο? γεγράφθω ό 
ΔΚΒ' εσται δή αντον διάμετρο? ή ΔΒ, δια το όρθήν είναι την προ? 

*5 τω Κ γωνίαν. και ενηρμόσθω ή ΒΑ εζαγώνον. και επει ή νπο 
των ΔΒΕ γωνία λ / εστίν ορθή?, και ή νπο των ΒΔΚ άρα λ' εστίν 
ορθή?' ή άρα Β Κ περιφέρεια £' εστίν τον όλον κνκλον. εστίν δε 
καϊ ή ΒΑ έκτον μερο? τον δλον κνκλον ή άρα ΒΑ περιφέρεια τή? 
ΒΚ περιφέρεια? ι εστίν, και έχει ή ΒΑ περιφέρεια προ? την ΒΚ 

2ο περιφερειαν μείζονα λόγον ήπερ ή ΒΑ ενθεΐα προ? την ΒΚ ενθεΐαν 
ή άρα ΒΑ ενθεΐα τή? Β Κ ευθεία? ελάσσων εστίν ή ι. καϊ εστίν 
αντή? διπλή ή ΒΔ' ή άρα ΒΔ τή? ΒΚ ελάσσων εστίν ή κ. ώ? δε 
ή ΒΔ προ? την Β Κ, ή ΑΒ προ? (την) ΒΤ, ώστε και ή ΑΒ τή? 
Β1 ελάσσων εστίν η κ* και εστίν η μεν ΑΒ το απόστημα ο 

25 απέχει ό ήλιο? άπο τή? γή?, ή δε ΒΤ το απόστημα ο απέχει 
ή σελήνη άπο τή? γή?' το άρα απόστημα ο απέχει ό ήλιο? άπο τή? 
γή? τον άποστήματο?, ον άπεχει ή σελήνη άπο τή? γή?, έλασσον 
εστίν ή κ. εδείχθη δε και μείζον ή ιη. 



I. HE] EHW 4, 5» 6• 8. ιη] δκτωκαιδεκαπλασίων W 8. το] om. W 

ΙΟ. ον] b W F Nizze, sed cf. 1. 3> Ρ• 37^ II. μείζον] μείζων W ιη] οκτω- 

καιδεκαπλάσιον W 12. κ] €ΐκοσιπ\άσιονν? τον] (τό) W Ι3• ΔΚΒ] 

ΔΒΚ W 1 5. τώ] τ 6 W Κ] Γ Vat. 16. λ' (bis)] τριακοστοί, W ιγ. ξ] 
εξακοστόν W 1 9, 21. ι] δεκαπλασίων W 22, 24• κ] εϊκοσιπΚασίων W 

22. ως] ώστε W 23. η ΑΒ (prius)] όντως η ΑΒ W (την)] ΟΙΏ. Vat. Paris. 

2366, leg. Paris. 2364, 2488 ιη. ον] ο WF Nizze, sed cf. 1. 3, p. 376 

28. κ] είκοσιπλάσιον W μείζον] μείζων W Vat. ιη] οκτωκαώεκαπλάσιον W 



OF THE SUN AND MOON 381 

But it was also proved that GE has to EH a ratio greater than 
that which 15 has to 2 ; 

therefore, ex aequali, FE has to EH a ratio greater than that 
which 36 has to 2, that is, than that which 18 has to 1 ; 
therefore FE is greater than 18 times EH. 

And FE is equal to BE ; 
therefore BE is also greater than 18 times EH; 
therefore Β Η is much greater than 18 times HE. 

But, as BH is to HE> so is A Β to BC, because of the similarity 
of the triangles ; 
therefore A Β is also greater than 18 times BC. 

And A Β is the distance of the sun from the earth, while CB is 
the distance of the moon from the earth ; therefore the distance of 
the sun from the earth is greater than 1 8 times the distance of the 
moon from the earth. 

Again, I say that it is also less than 20 times that distance. 

For let DK be drawn through D parallel to EB, and about the 
triangle DKB let the circle DKB be described ; then DB will be 
its diameter, because the angle at Κ is right. 

Let BL, the side of a hexagon, be fitted into the circle. 

Then, since the angle DBE is 1/30Λ of a right angle, the angle 
BDK is also i/30th of a right angle ; 
therefore the circumference Β Κ is i/6oth of the whole circle. 

But BL is also one sixth part of the whole circle. 

Therefore the circumference i?Z, is ten times the circumference BK. 

And the circumference BL has to the circumference Β Κ a ratio 
greater than that which the straight line BL has to the straight 
line Β Κ j 1 

therefore the straight line BL is less than ten times the straight 
line BK. 

And BD is double of BL ; 
therefore BD is less than 20 times BK. 

But, as BD is to BK, so is A Β to BC ; 
therefore A Β is also less than 20 times BC. 

And A Β is the distance of the sun from the earth, 
while BC is the distance of the moon from the earth ; 
therefore the distance of the sun from the earth is less than 20 times 
the distance of the moon from the earth. 

And it was before proved that it is greater than 1 8 times that 
distance. 

1 By the proposition proved in Ptolemy's Syntaxis, i, 10, pp. 43-4, ed. 
Heiberg. See, for his proof, the note on pp. 368-9, above. 



382 ON THE SIZES AND DISTANCES 



V • 



'Οταν 6 ήλιος εκλείπτ} όλος, τότε ό αύτος κώνο? περι- 
λαμβάνει τον τε ήλιον και την σελήνην, την κορνφήν 
έχων προς τ{} ημέτερα όψει, 

5 Έ7Γ€ί γαρ, εαν εκλείπχι 6 ήλιος, δι επιπρόσθεσιν της σελήνης 
εκλείπει, εμπίπτοι αν ό ήλιος εις τον κωνον τον περιλαμβάνοντα την 
σελήνην την κορυφήν Έχοντα προς ττ\ ημέτερα όψει. εμπίπτων δε 
ήτοι εναρμόσει εις αυτόν, η ύπεραίροι, ή ελλείποι* εί μεν ουν 
ύπεραίροι, ουκ (αν) εκλείποι όλος, άλλα, παραλλάττοι αυτόν το 

ίο ύπεραΐρον. ει δε ελλείποι, διαμενοι αν εκλελοιπως εν οσω διε£- 
εργεται το ελλεΐπον. όλος δε εκλείπει και ου διαμένει εκλελοιπώς- 
τούτο γαρ εκ της τηρήσεως φανερόν. ώστε οϋτ αν ύπεραίροι, οϋτε 
ελλείποι. εναρμόσει άρα εις τον κωνον, και περιληφθήσεται ύπο 
του κώνου του περιλαμβάνοντος την σελήνην την κορυφήν έχοντος 

1 5 προς ττ\ ημέτερα όψει. 

θ'. 

Ή του ηλίου διάμετρος της διαμέτρου της σελήνης 
μείζων μεν εστίν ή ιη, ελάσσων δε ή κ. 

Έστω γαρ ή μεν ημέτερα όψις προς τω Α, ηλίου δε κεντρον το 

20 Β, σελήνης δε κεντρον το Γ, όταν ό περιλαμβάνων κώνος τον τε 

ήλιον και την σελήνην την κορυφήν εχτ} προς ττ) ημέτερα όψει, 

τουτέστιν , όταν τα Α, Γ, Β σημεία επ' ευθείας ή, και εκβεβλήσθω 

δια της ΑΤΒ επίπεδον ποιήσει δή τομάς, εν μεν ταΐς σφαίραις 

Ι. η] om. Vat. 7• προ? τη ημέτερα ox/^ei] προς την ημίτίραν όψιν Vat. 

8. €ναρμόσ€ΐ] έναρμώσα Vat. eXX«7roi] eWdnei Paris. 2364 : έκλύποι F Paris. 
2488 9. ουκ (αν) εκλείποι] συν(κλΰποι Vat. p.m. : ουκ έκλύποι Vat. corr. 

(ουκ supra lineam scripto) F Paris. 2488 : ουσυνεκΚύποι Paris. 2366 : ουκ eVcXf ίπβι 
Paris. 2342, 2364 : ουκ iWefcei W IO. eXXeiVoi] eXXeiVei W Paris. 2364 : 

εκλείποι F Paris. 2342, 2488 13. ελλείποι] εκλείποι F Paris. 2488: ελλείπει 

Paris. 2364 

16. θ'] Η Vat. 18. ιη] οκτωκαώεκαπλασίων W κ] εϊκοσιπλασίων W 

21 ' *Χΐ)] %Χ ζ1 Vat. 



OF THE SUN AND MOON 383 

Proposition 8. 

When the sun is totally eclipsed, the sun and the moon are then 
comprehended by one and the same cone which has its vertex at 
our eye. 

For since, if the sun is eclipsed, it is eclipsed because the moon 
is in front of it, 

the sun must fall into the cone comprehending the moon and having 
its vertex at our eye. 

And, if it falls into it, either it will exactly fit into it, or it must 
overlap it or fall short of it. 

If now it should overlap it, the sun would not be totally eclipsed, 
but the portion which overlaps would be unobstructed. 1 

If, however, it should fall short, the sun would remain eclipsed 
for the time which it takes to pass through the portion by which 
it falls short. 

But it is in fact totally eclipsed and does not remain eclipsed : for 
this is manifest from observation. 2 

Hence it can neither overlap nor fall short ; 
therefore it will exactly fit into the cone and will be comprehended 
by the cone comprehending the moon and having its vertex at 
our eye. 

Proposition 9. 

The diameter of the sun is greater than 18 times \but less than 
20 times, the diameter of the moon. 

For let our eye be at A, let Β be the centre of the sun, and C the 
centre of the moon when the cone comprehending both the sun 
and the moon has its vertex at our eye, that is, when the points 
A, C, Β are in a straight line. 

Let a plane be carried through A CB ; 
this plane will cut the spheres in great circles and the cone in 
straight lines. 

1 Gr. παράλλάττοι. As in Euclid, i. 7 and iii. 24, παραλλάτταν means to 'fall 
beside ' or ' awry ', to ' miss ', to ' pass by without touching '. 

2 It is evident from this that Aristarchus had not observed the phenomenon 
of an annular eclipse of the sun. The first mention of annular eclipses on 
record appears to be that quoted by Simplicius (on De caelo, ii. 12, p. 505, 7-9, 
Heiberg) from Sosigenes, the teacher of Alexander Aphrodisiensis (end of 
second century A.D.). 



384 ON THE SIZES AND DISTANCES 

μεγίστου? κύκλους, kv δε τω κώνω ευθείας, ποιείτω ουν εν μεν ταΐς 
σφαίραις μέγιστους κύκλους τους ΖΗ, ΚΑΘ, εν δε τω κώνω 
ευθείας τα,ς ΑΖΘ, ΑΗΚ, και επεζεύχθωσαν αϊ ΓΗ, ΒΚ, εσται 




Fig. 27. 

δή, ως ή ΒΑ προς την AT, ή ΒΚ προς ΓΗ. ή δε ΒΑ τής ΑΓ 
5 εδείγθη μείζων μεν ή ιη, ελάσσων δε ή κ. κα\ ή ΒΚ άρα της ΓΗ 
μείζων μεν εστίν η ιη, ελάσσων δε ή κ. 



ι . 



Ό ήλιος προς την σελήνην μείζονα μεν λόγον έ'χ€ί ή 
ον τα εωλβ προς α, ελάσσονα δε ή ον τα η προς α. 

10 Εστω ή μεν του ηλίου διάμετρος ή Α, ή δε της σελήνης ή Β. 
ή Α άρα προς την Β μείζονα λόγον έχει ή ον τα ιη προς α, 
ελάσσονα δε ή ον τα κ προς α. και επειδή δ άπο τής Α κύβος προς 
τον άπο τής Β κύβον γ λόγον εγει ήπερ ή Α προς την Β, εγει δε 
και ή περί διάμετρον την Α σφαίρα προς την περί διάμετρον τήν Β 

ΐ5 σφαΐραν γ λόγον ήπερ ή Α προς τήν Β, εστίν άρα } ως ή περί 

διάμετρον τήν Α σφαίρα προς τήν περί διάμετρον τήν Β σφαΐραν, 

ούτως ό άπο τής Α κύβος προς τον άπο τής Β κύβον. ό δε άπο τής 

Α κύβος προς τον άπο τής Β κύβον μείζονα λόγον εγει ή ον τα 

εωλβ προς α, ελάσσονα δε ή ον τα η προς α, επειδή ή Α προς τήν 

2ο Β μείζονα λόγον εγει ή ον τα ιη προς α, ελάσσονα δε ή ον τα κ 
προς εν ώστε ό ήλιος προς τήν σελήνην μείζονα λόγον έχει ήπερ τα 
εωλβ προς α, ελάσσονα δε ή ον τα η προς α. 

ι. €νθ€ίας] W Paris. 2364: (νθεϊαν F Vat. Paris. 2366, 2488 μίρ] om. W 

2. τω κωνω] τοις κωνοις Vat. Paris. 2366, 2488 4• V ΒΚ\ (όντως) ή ΒΚ Nizze 

ΓΗ] τήν rHW 5. h »?] Ψ Vat. 6. tj ιη] ιη Vat. 

7• »"] θ Vat. 13, Ι5• y] sic Vat. pro τριπλασίονα : τριπλασίονα W 

14. προς την] προς την Β Vat. 21. ev] oW 



OF THE SUN AND MOON 385 

Let it cut the spheres in the great circles BG y KLH, and the 
cone in the straight lines AFH, AGK, 
and let CG, Β Κ be joined. 

Then, as Β A is to A C, so will Β Κ be to CG. 

But it was proved that Β A is greater than 18 times, but less 
than 20 times, A C. [P ro P• 7] 

Therefore Β Κ is also greater than 18 times, but less than 20 
times, CG. 

Proposition 10. 

The sun has to the moon a ratio greater than that which 5832 
has to 1, but less than that which 8000 has to 1. 

Let A be the diameter of the sun, Β that of the moon. 



Β — 

Fig. 28. 

Then A has to Β a ratio greater than that which 18 has to i, 
but less than that which 20 has to 1. 

Now, since the cube on A has to the cube on Β the ratio triplicate 
of that which A has to B, 

while the sphere about A as diameter also has to the sphere about 
Β as diameter the ratio triplicate of that which A has to B, 
therefore, as the sphere about A as diameter is to the sphere about 
Β as diameter, so is the cube on A to the cube on B. 

But the cube on A has to the cube on Β a ratio greater than 
that which 5832 has to 1, but less than that which 8000 has to 1, 
since A has to Β a ratio greater than that which 1 8 has to 1 , but 
less than that which 20 has to 1. 

Accordingly the sun has to the moon a ratio greater than that 
which 5832 has to i, but less than that which 8000 has to 1. 

1410 C C 



386 ON THE SIZES AND DISTANCES 

ια. 

Ή τής σελήνης διάμετρος τον αποστήματος, ον άπεχει 
το κεντρον της σελήνης άπο της ημετέρας όψεως, ελάσσων 
μεν εστίν ή όνο με , μείζων οε η λ . 

5 'Εστω γαρ ή μεν ημέτερα οψις προς τω Α, σελήνης δε κεντρον 
το Β, όταν ό περιλαμβάνων κώνος τον τ€ ήλιον και την σελήνην την 
κορνφήν βχτ} προς τή ημέτερα όψει, λέγω οτι γίγνεται τα δια της 
προτάσεως. 

'Έ,πεζεύγθω γαρ ή ΑΒ, και εκβεβλήσθω το δια της ΑΒ επίπεδον 

ίο ποιήσει δη τομην kv μεν ttj σφαίρα κύκλον, εν δ\ τω κωνω ευθείας, 
ποιείτω ου ν εν μεν ttj σφαίρα κνκλον τον ΓΕΑ, εν δε τω κωνω 
ευθείας τάς ΑΔ, AT, και επεζεύχθω ή ΒΓ, και διήχθω επί το Ε. 
φανερον δή εκ τον προδεδειγμενον οτι ή νπο των ΒΑΤ γωνία 
ημισείας ορθής εστί με • και κατά τα, αντα ή Β Τ της ΓΑ ελάσσων 

ΐ5 εστίν ή με . πολλω άρα ή ΒΓ της ΒΑ ελάσσων εστίν ή με 
μέρος, και εστί της Β Γ διπλή ή ΓΕ' ή Γ Ε άρα τής ΑΒ ελάσσων 
εστίν ή δύο με . και εστίν ή μεν ΓΕ ή τής σελήνης διάμετρος, ή 
δε ΒΑ το απόστημα ο απέχει το κεντρον τής σελήνης άπο τής 
ημετέρας όψεως' ή άρα διάμετρος τής σελήνης τον αποστήματος, ον 

so απέχει το κεντρον τής σελήνης άπο τής ημετέρας όψεως, ελάσσων 
εστίν ή όνο με . 

Λέγω δή οτι και μείζων εστίν ή ΓΕ τής ΒΑ ή λ μέρος, 
επεζεύχθω γαρ ή ΔΕ και ή Δ Γ, καϊ κεντρω μεν τω Α, διαστήματι 
δε τω ΑΓ, κύκλος γεγράφθω 6 ΓΔΖ, και ενηρμοσθω εις τον ΓΔΖ 

2$ κύκλον τή ΑΓ ΐση ή ΔΖ. και επει ορθή ή νπο των ΕΔΓ ορθή τή 

ι. ια'] I Vat. 2. ον] δ W F Nizze, sed nihil mutandum 4. με] τεσσαρα- 
κοστόπεμπτα W λ'] τριακοστόν W 6. περιλαμβάνων] παραλαμβάνων W 

7• ϊχΐΐ] *Χ" Vat. γίγνεται] γίνεται W δια] om. W Ι3• ^»] 

άπο W 14, Ι5• μ ( '] τεσσαρακοστόπεμπτον W Ι4• ΓΑ] ΒΑ W Ι5~ΐ6. 

πο\λω άρα . . . με' μέρος] om. W Paris. 2366 IJ. με'] τεσσαρακοστόπεμπτα W 

Ι9• ου] ο W F Nizze, sed cf. 1. 2 supra 2I. με] τεσσαρακοστόπεμπτα W 



OF THE SUN AND MOON 



3*7 



Proposition ii. 

The diameter of the moon is less than 2/^ths, but greater than 
i/$oth, of the distance of the centre of the moon from our eye. 

For let our eye be at Λ, and let Β be the centre of the moon 
when the cone comprehending both the sun and the moon has its 
vertex at our eye. 

I say that the above proposition is true. 

Let A Β be joined, and let the plane through A Β be drawn ; 
this plane will cut the sphere [i.e. the moon] in a circle and 
the cone in straight lines. 

Let it cut the sphere in the circle CED and the cone in the 
straight lines AD, AC; 
let BC be joined and produced to E. 

Then it is manifest from what has before been proved [Prop. 4] 
that the angle Β A C is i/45th part of half a right angle ; 
and, in the same way as 
before, BC is less than 
i/45th part of CA ; 
therefore BC is much less 
than i/45th part of Β A. 

And CE is double of 
BC; 

therefore CE is less than 
2/45ths of AB. 

Now CE is the diameter 
of the moon, 

while Β A is the distance 
of the centre of the moon 
from our eye. 

Therefore the diameter 
of the moon is less than 
2/45ths of the distance of the centre of the moon from our eye. 

I say next that CE is also greater than i/30th part of Β A. 

For let DE and DC be joined, and with centre A and distance 
A C let the circle CDFbe described ; let DE equal to A C be fitted 
into the circle CDF. 

C c 2 




Fig. 29. 



388 ON THE SIZES AND DISTANCES 

νπο των BTA εστίν ϊση, αλλά και ή νπο των ΒΑΤ τή νπο των 
ΘΓΒ εστίν ϊση, λοιπή άρα ή νπο των ΔΕΤ λοιπή τή νπο των 
ΘΒΓ εστίν ϊση' ίσογώνιον άρα εστίν το Τ ΔΕ τρίγωνον τω ΑΒΓ 
τριγώνω. εστίν άρα, ω? ή ΒΑ προς AT, όντως η ΕΤ προς ΤΔ- 

5 και εναλλάξ, ω? ή ΑΒ προ? ΤΕ, όντως ή AT προ? ΤΔ, τοντεστιν, 
η ΔΖ προ? ΤΔ. αλλ' €7rei πάλιν η νπο των ΔΑΤ γωνία με' μύρο? 
εστίν ορθή?, ή ΤΔ άρα περιφέρεια ρπ' μύρο? εστί τον κνκλον ή δε 
ΔΖ περιφέρεια έκτον μερο? εστίν τον όλον κνκλον ώστε ή ΤΔ 
περιφέρεια τή? ΔΖ περιφέρεια? λ / μερο? εστίν, και έχει ή ΤΔ 

ίο περιφέρεια, ελάσσων ονσα τή? ΔΖ περιφέρεια?, προ? αντήν την ΔΖ 
περιφερειαν ελάσσονα λόγον ήπερ ή ΤΔ ενθεΐα προ? την ΖΔ 
ενθεΐαν ή άρα ΤΔ ενθεΐα τή? ΔΖ μείζων εστίν η λ', ϊση δε ή 
Ζ Δ τή AT' ή άρα Δ Γ τή? ΓΑ μείζων εστίν η λ', ώστε και ή ΤΕ 
τή? ΒΑ μείζων εστίν η Χ. εδείχθη δε καΐ ελάσσων ονσα ή δνο με'. 

15 ρ 

Ή διάμετρο? τον διορίζοντο? εν τή σελήνη το τε 

σκιερον και το λαμπρον τή? διαμετρον τή? σελήνη? 

ελάσσων μεν εστί, μείζονα δε λόγον έχει προ? αντήν ή 

ον τα πθ προ? ς. 

2ο Έστω γαρ ή μεν ημέτερα οψι? προ? τω Α, σελήνη? δε κεντρον 
το Β, όταν 6 περιλαμβάνων κώνο? τον τε ήλιον και την σελήνην την 
κορνφήν εχη προ? τή ημέτερα όψει, και επεζενγθω ή ΑΒ, και 
εκβεβλήσθω δια τή? ΑΒ επίπεδον ποιήσει δη τομά? εν μεν τή 
σφαίρα κνκλον, εν δε τω κώνω ενθεία?. ποιείτω (εν μεν τή σφαίρα 

25 κνκλον τον ΔΕ Τ, εν δε τω κώνω ενθεία?) τα? ΑΔ, AT, Τ Δ. 

2. ΘΓΒ] ΕΓΔ W 6. μ('] τ^σσαρακοστόπ^μπτον W 9• λ'] τριακοστον W 

ΙΟ. περιφίριια] add. προς την ΔΖ περιφερειαν Vat. II. λόγον] λόγον e\ei Vat. 

12, 14. λ'] τριακοστον W 1 2-13- 'ίση δί ή . . . η λ'] Om. W Ι φ /*«'] Tea- 

σαρακοστόπεμπτα W 

Ι5• Φ'] ΙΑ Vat. 21. περιλαμβάνων] παραλαμβάνουν W 24-25. (Jv Ρ* ν • • • 

ev6das) supplevit W : om. codd. 



OF THE SUN AND MOON 389 

Then, since the right angle ED C is equal to the right angle BCA , 
while the angle Β A C is also equal to the angle HCB, 
therefore the remaining angle DEC is equal to the remaining 
angle HBC. 

Therefore the triangle CD Ε is equiangular with the triangle ABC 

Therefore, as Β A is to A C, so is EC to CD ; 
and, alternately, as A Β is to CE, so is AC to CD, that is, DF 
to CD. 

But again, since the angle DA C is i/45th part of a right angle, 
the circumference CD is 1/1 80th part of the circle ; 
and the circumference DF is one sixth part of the whole circle ; 
thus the circumference CD is i/30th part of the circumference DF. 

And the circumference CD, being less than the circumference 
DF, has to the circumference DF itself a ratio less than that which 
the straight line CD has to the straight line FD. 1 

Therefore the straight line CD is greater than i/30th of DF. 

But FD is equal to AC; 
therefore DC is greater than i/30th of CA, 
so that CE is also greater than i/30th οί Β A [see above]. 

But it was before proved to be also less than 2/45ths. 

Proposition 12. 

The diameter of the circle which divides the dark and the bright 
portions in the moon is less than the diameter of the moon, but 
has to it a ratio greater than that which 89 has to 90. 

For let our eye be at A , and let Β be the centre of the moon 
when the cone comprehending both the sun and the moon has its 
vertex at our eye ; 

let A Β be joined, and let a plane be carried through AB; this 
plane will cut the sphere [i.e. the moon] in a circle and the cone in 
straight lines. 

Let it cut the sphere in the circle DEC and the cone in the 
straight lines AD, AC, CD. 

1 For, by the proposition proved by Ptolemy (see note on Prop. 4, above), 
FD: DC< (arc FD) : (arc DC), 
and, by inversion, (arc CD) : (arc DF) < CD : DF. 



39o 



ON THE SIZES AND DISTANCES 



η ΓΑ άρα διάμετρο? εστί τον κύκλου τον διορίζοντος εν τή σελήντ) 

το σκιερον και το λαμπρόν. λέγω δη οτι ή ΓΑ της διαμίτρον της 

σελήνης ελάσσων μεν εστί, μείζονα δε λόγον έχει (προς αντην) η ον 

τα πθ προς ς. 
5 "Οτι μεν ονν ή ΓΑ ελάσσων εστί της διαμίτρον της σελήνης, 

φανερόν. λέγω δη οτι και μείζονα λόγον έχει (προς αυτήν) ή ον 

τα, πθ προς ς. 

"Ή,χθω γαρ δια τον Β τή ΓΑ παράλληλος ή ΖΗ, και επε- 

ζενχθω ή Β Γ. εσται δή πάλιν κατά τα αντα. ή νπο των ΑΛΓ 
ίο γωνία ορθής με' μέρος, ή δε νπο των ΒΑΓ ορθής ς' μέρος. και 



c D 




F C 



Fig. 30. 

εστίν ή νπο των ΒΑ Γ γωνία ϊση τή νπο των ΓΒΖ• και ή νπο των 
ΓΒΖ άρα γωνία ορθής εστίν ς', τοντεστιν, τής νπο των ΖΒΕ 
γωνίας ς', ώστε και ή ΓΖ περιφέρεια τής ΖΓΕ περιφερείας εστίν 
ς', ή ΓΕ άρα περιφέρεια προς την ΕΓΖ περιφερειαν λόγον έχει 

ι 5 ον τα πθ προς ς. και εστί τής ΓΕ β ή ΑΕΓ, τής δε ΕΓΖ β ή 
ΗΕΖ' ή άρα ΑΕΓ περιφέρεια προς την ΗΕΖ περιφερειαν λόγον 
έχει ον τα πθ προς ς. και έχει ή Α Γ ευθεία προς (την) ΗΖ 
ενθεΐαν μείζονα λόγον ήπερ ή ΑΕΓ περιφέρεια προς την ΗΕΖ 
περιφερειαν και ή Α Γ άρα ενθεΐα προς την Η Ζ ενθεΐαν μείζονα 

^ ο λόγον έχει ή ον τα πθ προς ς. 

2. δη] δε W 3> 6. (προς αντην)] addidi ΙΟ. /ie'J τ^σσαρακοστό- 

π(μπτον W των] τον Vat. ΙΟ, 12, Ι4• <}'] ένν^νηκοστόν W 13. γωνίας] γωνίας, 
■γωνίας W 1$. β bis] διπλή W \η. και ?χ«] εχβι post \6yov (1. 18) posuit W 



OF THE SUN AND MOON 391 

Therefore CD is a diameter of the circle which divides the dark 
and the bright portions in the moon. 

I say that CD is less than the diameter of the moon, but has to it 
a ratio greater than that which 89 has to 90. 

Now, that CD is less than the diameter of the moon is manifest. 

I say, then, that it also has to it a ratio greater than that which 
89 has to 90. 

For let FG be drawn through Β parallel to CD, 
and let BC be joined. 

Then again, in the same way as before, the angle DA C will be 
1/45 th part of a right angle, 

and the angle Β A C will be i/cjoth part of a right angle ; 
but the angle Β AC is equal to the angle CBF\ 
therefore the angle CBF is also i/90th of a right angle, 
that is, i/90th of the angle FBE ; 

so that the circumference CF is also i/90th of the circumference 
FCE. 

Therefore the circumference CE has to the circumference ECF 
the ratio which 89 has to 90. 

Now DEC is double of CE, and GEF double of ECF; 
therefore the circumference DEC has to the circumference GEF 
the ratio which 89 has to 90. 

And the straight line DC has to the straight line GF a ratio 
greater than that which the circumference DEC has to the 
circumference GEF. 1 

Therefore also the straight line DC has to the straight line GF 
a ratio greater than that which 89 has to 90. 

1 By the proposition quoted from Ptolemy, i. 10, pp. 43-4, ed. Heiberg. See 
note on Props. 4 and II, above. 



392 ON THE SIZES AND DISTANCES 

iy\ 
Ή νποτείνονσα ενθεΐα νπο την άπολαμβανομίνην εν 
τω σκιάσματι τή? γή? περιφίρειαν τον κύκλου, καθ' ον 
φέρεται τα άκρα της διαμίτρον τον διορίζοντος kv τή 

5 σελήνη το τε σκιερον καϊ το λαμπρόν, τη? μεν διαμέτρου 
της σελήνη? ελάσσων εστίν η διπλή, μείζονα δε λόγον 
£'χ€* προ? αντην η ον τα. πη προ? με, τη? δε τον ήλίον 
διαμίτρον ελάσσων μεν εστίν ή ενατον μερο?, μείζονα 
δε λόγον έχει προ? αντην η ον κβ προ? σκε, προ? δε την 

ίο άπο τον κεντρον τον ήλίον ήγμίνην προ? ορθά? τω ά£ονι, 
σνμβάλλουσαν δε ταΐ? τον κώνον πλενραΐ?, μείζονα λόγον 

α 

έχει η ον τά ~*)οθ προ? Μρκε. 

ν Εστω yap ήλίον μεν κεντρον προ? τω Α, γη? δε κεντρον το Β, 
σελήνη? δε το Γ, τελεία? ονση? τη? εκλείψεω? και πρώτω? δλη? 

ΐ5 εμπεπτωκνία? ει? το τη? γη? σκίασμα, και εκβεβλήσθω δια, των Α, 
Β, Γ επίπεδον ποιήσει δη τομα? εν μεν ταΐ? σφαίραι? κύκλον?, εν 
δε τω κώνω ευθεία? τω περιλαμβάνοντι τον τε ήλιον και την γήν. 
ποιείτω εν μεν ταΐ? σφαίραι? μεγίστον? κνκλον? τον? ΔΕΖ, ΗΘΚ, 
ΑΜΝ, εν δε τω σκιάσματι τή? γη? κνκλον, καθ' ον φέρεται τά άκρα 

so τή? διαμίτρον τον διορίζοντο? εν τή σελήνη το τε σκιερον και το 
λαμπρόν, τον UAN, εν δε τω κώνω ευθεία? τά? AHa f ΖΚΝ' 
άξων δε έστω ό ΑΒΑ. φανερον δή δτι ό ΑΒΑ άξων εφάπτεται 
τον ΑΜΝ κνκλον, διά το το σκίασμα τή? γή? σεληνων είναι δύο, καϊ 
δίχα διαιρεΐσθαι την Ν AS περιφίρειαν νπο τον ΑΒΑ άξονο?, και 

25 ετι την σελήνην πρώτω? εμπεπτωκεναι ει? το τή? γή? σκίασμα. 
επεζενχθωσαν δή αϊ ΗΝ, ΝΔ, ΒΝ, ΛΒ. ή ΑΝ άρα εστίν ή 
διάμετρο? τον διορίζοντο? εν τή σελήνη τό τε σκιερον και το 
λαμπρόν, και ή ΒΝ εφάπτεται τον ΑΝ Ο Μ κύκλον, διά το είναι το 
Β προ? τή ημέτερα όψει, και την ΑΝ διάμετρον τον διορίζοντο? εν 

3ο τή σελήνη τό τε σκιερον και το λαμπρόν. και επει αϊ Ά Α, ΑΝ 

I. iy'] IB Vat. 2. την] om. W 7. με] τά με W 8. ενατον] εννατον W 

14. εκλείψεως] εκλίψεως Vat. 1 6. δή] δε W 1 8. tovs] om. W 1 9. κύκλοι/] 

κύκλων Vat. 22. ό 2°] ή W εφάπτεται] εφάπτηται W 25. πρωτως] πρωτως 
{ολην) Nizze 28-30. και ή ΒΝ εφάπτεται . . . λαμπρόν] om. soli, ut videtur,codd. 
Savilianus et Paris. 2342 28. εφάπτεται] ευθεία εφάπτεται W ΛΝΟΜ] F 

Paris. 2364, 2488 : AN ON Vat. : ΛΜΝ W 28-9. είναι τό Β] τό Β σημεΐον 

είναι W, qui lacunam post 1. 28 λαμπρόν ope versionis Commandini expleverat 
29 $4χμ€τρον] την διάμετρον είναι W 



OF THE SUN AND MOON 393 



Proposition 13. 

The straight line subtending the portion intercepted within the 
earth's shadow of the circumference of the circle in which 
the extremities of the diameter of the circle dividing the dark 
and the bright portions in the moon move is less than double 
of the diameter of the moon, but has to it a ratio greater than 
that which 88 has to 45 ; and it is less than i/glh part of the 
diameter of the sun, but has to it a ratio greater than that which 
22 has to 22 5. But it has to the straight line drawn from the 
centre of the sun at right angles to the axis and meeting the sides 
of the cone a ratio greater than that which 979 has to 101 25. 

For let the centre of the sun be at A, let Β be the centre of the 
earth, and C the centre of the moon when the eclipse first becomes 
total through the moon having• fallen wholly within the earth's 
shadow. 

Let a plane be carried through A, B, C ; 
this plane will cut the spheres in circles and the cone comprehending 
both the sun and the earth in straight lines. 

Let it cut the spheres in the great circles DEF, GHK, LMN, 
the earth's shadow in the circle OLN in which the extremities of 
the diameter of the circle dividing the dark and the bright portions 
in the moon move, and the cone in the straight lines DGO, FKN. 

Let ABL be the axis. 

Then it is manifest that the axis ABL touches the circle LMN, 
because the shadow of the earth is of two moon-breadths, [Hyp. 5] 
that the circumference NLO is bisected by the axis ABL, 
and further that the moon has for the first time fallen within the 
earth's shadow. 

Let ON, NL, BN, LO be joined. 

Therefore LN is the diameter of the circle dividing the dark 
and the bright portions in the moon. 

And Β Ν touches the circle LNPM, 
because the point Β is at our eye, and LN is the diameter of the 
circle dividing the dark and the bright portions in the moon. 



394 



ON THE SIZES AND DISTANCES 



ϊσαι είσίν, διπλασίονες άρα είσι της ΛΝ, ώστε ή ΗΝ της 

ελασσων εστίν η οι- 

πλή. επεζενγβωσαν 

δή αϊ ΛΓ, ΓΝ, και 
5 διήχθω ή ΛΓ επι το 

Ο- πολλω άρα ή &Ν 

της ΛΟ ελάσσων εσ- 

τιν η ρ. και επει η 

Γ Λ κάθετος 'εστίν επι 
ίο την ΒΛ, παράλληλος 

αρα εστίν ττ) aJS* ίση 

άρα εστίν ή νπο των 

ΛαΝ ττ) νπο των 

ΓΛΝ γωνία, και εσ- 
ΐ5 τιν ϊση μεν ή Ν Λ τ§ 

Λ3,ήδεΛΓττ)ΓΝ> 

δμοιον άρα εστίν το 

ΈΝΑ τρίγωνον τω 

ΛΝΓτριγώνω• εστίν 
2ο άρα,ώςήαΝ προς την 

Ν Λ , όντως ή Ν Λ προς 

τήνΛΓ. άλλ'ήΝΛ 

προς την ΛΓ μείζονα 

λογον έχει η ον τα πυ 
25 προς τα, με, τοντεστι, 

το άπο ΝΛ προς το 

άπο ΛΓ μείζονα λό- 

γον έχει ήπερ τα 

ζ^ίκα προς τα βκε' 

30 και το απο Λ1> αρα 

προς το άπο Ν Λ μεί- 
ζονα λογον εγει ήπερ 

τα ζ^ίκαπροςτά βκε, _. 

/• r Γ ' Fig. 31. 

καιή αΝπροςτηνΛΟ 

8. β] διττλί} W 3θ• ΗΑ Τ ] τη* ΗΝ W 3ΐ • ΝΛ] rr,s ΝΛ W 



ΛΝ 




OF THE SUN AND MOON 395 

Now, since OL, LN are equal, their sum is double of LN, so 
that ON is less than double of LN. 

Let LC, CW be joined ; and let LC be carried through to P. 

Therefore ON is much less than double of LP. 

And, since CL is perpendicular to BL, 
therefore it is parallel to ON. 

Therefore the angle LON is equal to the angle CLN. 

And NL is equal to L 0, and LC to CN\ 
therefore the triangle ONL is similar to the triangle LNC ; 
therefore, as ON is to NL, so is NL to LC. 

But NL has to LC a ratio greater than that which 89 has to 45 ; l 
that is, the square on NL has to the square on LC a ratio greater 
than that which 7921 has to 2025. 

Therefore the square on ON also has to the square on NL 
a ratio greater than that which 7921 has to 2025, 
and (therefore) ON has to LP a ratio greater than that which 792 1 
has to 4050. 2 



1 For NL : LP > 89 : 90, by the preceding proposition. 

2 We have ON : NL = NL : LC ; 
therefore ON : LC = (sq. on ON) : (sq. on NL) 

> 7921 : 2025, 
whence ON : LP > 7921 : 4050. 



396 ON THE SIZES AND DISTANCES 

μείζονα λόγον έχει ήπερ τα ζ^ίκα προς δν. έχει δε και τα, ζ^ίκα 
προς δν μείζονα λόγον ήπερ τα πη προ? με' ή Να άρα προς ΑΟ 
μείζονα λόγον έχει ή ον τα, πη προς τα, με. ή άρα νποτείνονσα νπο 
την άπολαμβανομένην εν τω σκιάσματι της γης περιφέρειαν τον 
5 κύκλου, καθ'' ον φέρεται τα, άκρα της διαμέτρου τον διορίζοντος kv ttj 
σελήνη τό τε σκιερον και το λαμπρόν, της διαμετρον της σελήνης 
ελάσσων μεν εστίν η β, μείζονα δε λόγον έχει (προς αντην) ή ον τα, 
πη προς με. 

Των αντων υποκείμενων, ήχθω άπο τον Α τη ΑΒ προς όρθας 

ίο η ΠΑΡ' λέγω οτι ή UN της διαμετρον τον ήλίον ελάσσων μεν 

εστίν ή θ' μέρος, μείζονα δε λόγον έχει προς αντην η ον τα κβ προς 

τα σκε, προς δε την UP μείζονα λόγον έχει ή ον τα ~^ίοθ προς 

α 

Μρκε. έπει γαρ εδείχθη ή UN της διαμετρον της σελήνης ελάσσων 
ούσα ή β, ή δε διάμετρος της σελήνης της διαμετρον τον ήλίον 

1 5 ελάσσων εστίν ή ιη' μέρος, ή άρα UN της διαμετρον τον ήλιου 
ελάσσων εστίν ή θ μέρος, πάλιν έπει ή UN προς την διαμετρον 
της σελήνης μείζονα λόγον έχει ή ον τα, πη προς τα, με, ή δε 
διάμετρος της σελήνης προς την του ηλίου διαμετρον μείζονα λόγον 
έχει ή ον τα, με προς ~V έπει γαρ ή της σελήνης διάμετρος προς 

2θ την του ήλιου μείζονα λόγον έχει ή $>ν α προς κ, και πάντα 
τεσσαρακοντάκις και πεντάκις' ε£ει άρα ή UN προς την διαμετρον 
του ηλίου μείζονα λόγον ή ον τά πη προς τα, ~^, τουτέστιν, ή ον 
τά κ β προς τα, σκε. ήχθωσαν δη άπο του Β του Α Ε κύκλου 
έφαπτόμεναι αϊ ΒΎΧ, ΒΦΤ, και έπεζεύχθω ή ΎΦ και ή ΤΑ. 

25 έσται δή, ως ή διάμετρος τον διορίζοντος εν τη σελήνη τό τε σκιερον 
και το λαμπρον προς την διαμετρον της σελήνης, όντως ή ΎΦ προς 
την διαμετρον τον ήλίον, δια το τον αντον κωνον περιλαμβάνειν τον 



2. η Ν% άρα] άρα η Ν& Vat. 3• τ " Μ 6 ] Μ 6 W νποτείνονσα] 

άποτείνονσα Vat. 7• β] διπλή W (προς αντην)] addidi II, 1 6. θ'] 

ίννατον W 12. σκε] κε Vat. *X el ] *Χ (1 προς αντην W Ι4• β] 

διπλασίων W 15•^ ιη'] ιη' Vat. 1 7 '. τά με] με W 20. α] τό α W 

21. τεσσαρακοντάκις και πεντάκις] τεσσαρακοντακαιπεντάκις W 22. τά ~^)] 

~* W 23- ΔΕ] ΔΕΖ W 




OF THE SUN AND MOON 397 

But 792 1 also has to 4050 a ratio greater than that which 88 has 

to 45 J 1 

therefore NO has to LP a ratio greater than that which 88 has 

to 45• 

Therefore the straight line which subtends the portion inter- 
cepted within the earth's shadow of the circumference of the circle 
in which the extremities of the diameter of the circle dividing the 
dark and the bright portions in the moon move is less than double 
of the diameter of the moon, but has to it a ratio greater than that 
which 88 has to 45. 

The same suppositions being made, let QAR be drawn ftom A 
at right angles to AB. 

I say that ON is less than i/cjth part of the diameter of the sun, 
but has to it a ratio greater than that which 22 has to 225, and has 
to QR a ratio greater than that which 979 has to 10125. 

For, since it was proved that ON is less than double of the 
diameter of the moon, 

while the diameter of the moon is less than i/i8th part of the diameter 
of the sun, [Prop. 9] 

therefore ON is less than i/9th part of the diameter of the sun. 

Again, since ON has to the diameter of the moon a ratio greater 
than that which 88 has to 45, 

while the diameter of the moon has to the diameter of the sun 
a ratio greater than that which 45 has to 900 : 

for, since the diameter of the moon has to the diameter of the sun 
a ratio greater than that which 1 has to 20, we have only to 
multiply throughout by 45 : 

therefore (ex aequali) ON has to the diameter of the sun a ratio 
greater than that which 88 has to 900, that is, than that which 22 
has to 225. 

Now let BUS, BVT be drawn from Β touching the circle DE : 
and let UV, UA be joined. 

Then, as the diameter of the circle dividing the dark and the bright 
portions in the moon is to the diameter of the moon, so is UV 
to the diameter of the sun, because the sun and the moon are 

1 If we develop JfyJ as a continued fraction, we easily obtain the approximation 
1 + ΐ+ 2T+ 2 i which is in fact ff . See the similar case in Prop. 15, p. 407, and 
the observation thereon, p. 336 ad fin. 



398 ON THE SIZES AND DISTANCES 

τε ήλιον και την σελήνην την κορνφήν Έχοντα προς ttj ημέτερα όψει. 
ή δε διάμετρος τον διορίζοντος εν ττ) σελήντ) το τε σκιερον και το 
λαμπρον προς την διάμβτρον της σελήνης μείζονα λόγον έχει ή ον τα. 
πθ προς τα, ς* και ή ΤΦ άρα προς την τον ήλίον διάμβτρον μείζονα 

5 λόγον έχει η ον τά πθ προς ς* και ή XT άρα προς την ΤΑ μείζονα 
λόγον έχει η ον τα πθ προς ς. ώς δε ή XT προς την ΤΑ, όντως η 
ΤΑ προς την ΑΧ, δια το παραλλήλονς είναι τα,ς ΧΑ, ΤΧ• και η 
ΤΑ άρα προς την ΑΧ μείζονα λόγον έχει ή ον τα, πθ προς τα, ς• 
πολλω άρα ή ΤΑ προς την Α Ρ μείζονα λόγον έχει ή ον τα, πθ προς 

ίο τα, ς. και τα, β- ή άρα διάμετρος τον ήλίον προς την ΠΡ μείζονα 
λόγον έχει ή ον τα, πθ προς τα, ς. εδείχθη δε και ή ΗΝ προς την 
διάμετρον τον ήλίον μείζονα λόγον εχονσα ή ον τα κβ προς τα, σκε. 
δι 'ίσον πολλω άρα ή HIV προς την ΠΡ μείζονα λόγον έχει ή (ον) 

ό σννηγμενος εκ τε των κβ και πθ προς τον εκ των ς και σκε, 

β 
1 5 τοντεστιν, τα, α ^νη προς τα, Μσν και τα, ή μίση, τοντ εστίν, τα 

α 

~*ϊοθ npbs τά Μρκί. 



ι8'. 

Ή άπο τον κεντρον της γης επι το κεντρον της σελήνης 

επιζενγννμενη εύθεΐα προς την ευθείαν, ην απολαμβάνει 

so άπο τον άξονος προς τω κεντρω της σελήνης ή νπο την 

εν τω σκιάσματι της γης νποτείνονσα εύθεΐα, μείζονα 

λόγον *χ€ΐ ή ον τα, χοε προς α. 

"Εστω τό αντο σχήμα τω πρότερον, και ή σελήνη όντως έστω 
ώστε το κεντρον αντής είναι επι τον άξονος τον κώνον τον περί- 



4• τά ς] ς W ΙΟ. β] διπλάσια W 12. τά σκι] σκ( W 

β 
Ι5• Μσν] Μ/3, σν W ΐ6. τά Mp/ce] Μα. ρκ( W \η. ιδ'] ΙΓ Vat. 



OF THE SUN AND MOON 399 

comprehended by one and the same cone having its vertex at 
our eye. 1 

But the diameter of the circle dividing the dark and the bright 
portions in the moon has to the diameter of the moon a ratio 
greater than that which 89 has to 90 ; [Prop. 1 2] 

therefore UV also has to the diameter of the sun a ratio greater 
than that which 89 has to 90. 

Therefore WU also has to UA a ratio greater than that which 
89 has to 90. 

But, as WU is to UA, so is UA to AS, because SA, UWare 
parallel ; 

therefore UA also has to AS a ratio greater than that which 89 
has to 90; 

therefore UA has to AR a ratio much greater than that which 
89 has to 90. 

The same is true of the doubles ; 
therefore the diameter of the sun has to QR a ratio greater than 
that which 89 has to 90. 

But it was proved [above] that ON has to the diameter of the 
sun a ratio greater than that which 22 has to 225 ; 
therefore, ex aequali, ON has to QR a ratio much greater than 
that which the product of 22 and 89 has to the product of 90 and 
225, that is, 1958 to 20250, or, if the halves be taken, 979 to 
10125. 

Proposition 14. 

The straight line joined from the centre of the earth to the 
centre of the moon has to the straight line cut off from the axis 
towards the centre of the moon by the straight line subtending 
the {circumference) within the earth's shadow a ratio greater 
than that which 675 has to 1. 

For let the same figure be drawn as before ; 
and let the moon be so placed that its centre is on the axis of the 
cone comprehending both the sun and the earth ; 

1 The proof, which is given by Commandinus, is obvious. The fact cannot 
be seen from our figure, which, owing to exigency of space, could not be drawn 
so as to make the angles LBN, £/Z?F equal. 



400 



ON THE SIZES AND DISTANCES 



λαμβάνοντος τον τε ήλων και την γήν, καϊ έστω το Γ, μέγιστο? δε 
των kv τη σφαίρα κύκλος 6 ΠΟΜ kv τω αύτω επιπεδω ων αύτοΐς, 
και επεζεύχθω ή ΜΟ- ή ΜΟ άρα διάμετρος εστί τον διορίζοντος kv 
τη σελήνη τό τε σκιερον και το λαμπρόν. επεζεύχθωσαν δη αϊ 
5 MB, BO, Λα, αΒ, ΜΓ• εφάπτονται άρα του ΜΟΠ κύκλου αϊ 
MB, BO, δια το την Ο Μ διάμετρον είναι του διορίζοντος εν τη 




Fig. 32. 

σελήνη το σκιερον και το λαμπρόν. καϊ επει ϊση εστίν η ΕΛ τη 
Μ0• εκατερα γαρ αυτών διάμετρος εστί του διορίζοντος εν τη σελήνη 
το τε σκιερον και το λαμπρόν ίση άρα και ή αΜΛ περιφέρεια τη 
ίο ΜΑΟ περιφέρεια, και ή αΜ άρα ϊση εστίν τη ΑΟ. αλλ* ή ΑΟ 
ττ} AM ΐση εστίν καϊ ή αΜ άρα ίση εστίν τη AM. εστί δε και 
ή αΒ ΐση τη ΒΑ, δια το το Β σημεΐον κεντρον είναι της γης, και 

2. εν rjj σφαίρα] iv τη της σελήνης σφαίρα Nizze, suadente F, qui lectionem 
cod. Parisiensis 2488 σελήνη ante σφαίρα in σελήνης correxit ; mallem ε ν τη σελήνη 
pro εν τη σφαίρα, sed cf. 1. 14, p. 364 ; 1. io, p. 386 ; 1. 24, p. 388 4. δ/7] δε W 

7• το σκιερον] τό τε σκιερον W εστίν] om. W 



OF THE SUN AND MOON 401 

let its centre be C, and let the great circle QPM in the sphere 
[i. e. the moon] be in the same plane with the rest of the 
figure. 1 

Let MP be joined ; 
therefore MP is a diameter of 
the circle which divides the dark 
and the bright portions in the 
moon. 

Let MB, BP, LO, OB, MC be 
joined. 

Therefore MB, BP touch the 
circle MPQ, 

because PM is a diameter of the 
circle which divides the dark and 
the bright portions in the moon. 

And, since OL is equal to MP — 
for each of them is a diameter of 
the circle which divides the dark 
and the bright portions in the 
moon — 

therefore the circumference OML 
is equal to the circumference 
MLP\ 
therefore Ο Μ is also equal to LP. 

But LP is equal to LM ; 

therefore OM is also equal to LM. 

^ Fig. 33. 

And OB is equal to BL, 
because the point Β is the centre of the earth, and the earth has 




1 Literally 'in the same plane with them'* (avrols), which no doubt means the 
axis and the sections of the sun and moon made by the plane originally assumed, 
which also contains the circle in which the diameter of the ' dividing circle ' in 
the moon moves while the moon is passing through the earth's shadow. 
1410 D d 



4θ2 ON THE SIZES AND DISTANCES 

(την γήν) σημείου και κέντρου λόγον εχειν προ? την τη? σελήνη? 
σφαΐραν, και τον ΜΟΠ κύκλον kv τω αύτω επιπεδω ειναί' ή άρα 
Β Μ κάθετο? εστίν επι την %Λ. εστίν δε καϊ ή ΓΜ κάθετο? επι 
την ΒΜ' παράλληλο? άρα εστίν ή ΓΜ τή ΈΑ. εστί δε καϊ ή 
5 -5Έ rfj MP παράλληλο?' ομοιον άρα εστί το ABS τρίγωνον τω 
ΜΡΓ τριγώνω' εστίν apa f ω? ή Σ& προ? την MP, οϋτω? η Σ Λ προ? 
την ΡΓ. αλλ* ή ΣΗ τη? MP εστίν ελάσσων ή β, επει καϊ η UN 
τή? ΜΟ ελάσσων εστίν ή β' και ή ΧΑ άρα τη? ΤΡ ελάσσων 
εστίν η β• ώστε ή HP τη? ΡΓ πολλω ελάσσων εστίν η β. ή ΧΓ 

ίο άρα τή? ΓΡ ελάσσων εστίν η τριπλασίων' ή ΓΡ άρα προ? την ΓΧ 
μείζονα λόγον έχει ή ον α προ? γ. καϊ επεί εστίν, ω? η ΒΓ προ? 
ΓΜ, ούτω? η Γ Μ προ? την ΓΡ, ή δε Β Γ προ? την ΓΜ μείζονα 
λόγον έχει ή ον με προ? α, και ή ΓΜ άρα προ? ΓΡ μείζονα λόγον 
έχει η ον με προ? α. έχει όε και η 1 Ρ προ? την 1 2, μείζονα 

1 5 λόγον ή ον α προ? γ- δι 'ίσον άρα ή ΓΜ προ? την ΓΧ μείζονα 
λόγον έχει ή ον με προ? γ, τουτέστιν, (rj) ον ιε προ? α. εδείχθη δε 
και η ΒΓ προ? την ΓΜ μείζονα λόγον έχουσα η ον με προ? α• 
οι ίσου αρα η Β ι προ? την 12, μείζονα λογον έχει η ον τα χοε 
προ? α. 

2θ *€ • 

Ή του ηλίου διάμετρο? προ? την τή? γή? διάμετρον 
μείζονα λόγον έχει ή ον τα ιθ προ? γ, ελάσσονα δε η ον 
τα μγ προ? τα <Γ• 

Έστω γαρ ηλίου μεν κεντρον το Α, γή? δε κεντρον το Β, σελήνης 

2^ δε κεντρον το Γ, τελεία? οϋση? τή? εκλείψεω?, τουτέστιν, ΐνα τα, 

Α, Β, Γ επ' ευθεία? ff, και εκβεβλήσθω δια του άξονο? επίπεδον, 

Ι. {την yrjv)] haec verba prorsus necessaria solus habet Paris. 2364 
5. ΛΒΖ) ΛΖΒ W 7, 8, 9. β] διπλασίων W 12. την ΓΡ] ΓΡ W 13- α] 

μίαν Vat. Paris. 2366, 2488 : ϊν Paris. 2342, 2364 15• ΓΜ] ΜΓ W τήΑ 

om. W 16. de] δη W 

2o. ie'] ΙΔ Vat. 23. τα$] ς W 



OF THE SUN AND MOON 403 

the relation of a point and centre to the sphere in which the moon 
moves [Hyp. 2], while the circle MPQ is in the same plane ; 
therefore Β Μ is perpendicular to OL. 

But CM is also perpendicular to Β Μ ; 
therefore CM is parallel to OL. 

And SO is also parallel to MR ; 
therefore the triangle LOS is similar to the triangle MRC. 
Therefore, as SO is to MR, so is SL to RC. 
But SO is less than double οϊ MR, 
since ON is also less than double of MP; [Prop. 13] 

therefore SL is also less than double of CR, 
so that SR is much less than double of R C. 

Therefore SC is less than triple of CR ; 
therefore CR has to CS a ratio greater than that which 1 has to 3. 

And since, as BC is to CM, so is CM to CR, 
while BC has to CM a ratio greater than that which 45 has to 1, 

[see Prop. 11] 
therefore CM also has to CR a ratio greater than that which 
45 has to 1. 

But CR also has to CS a ratio greater than that which 1 
has to 3 ; 

therefore, ex aequali, CM has to CS a ratio greater than that 
which 45 has to 3, that is, than that which 15 has to 1. 

And it was proved that BC has to CM a ratio greater than that 
which 45 has to 1 ; 

therefore, ex aequali, BC has to CS a ratio greater than that which 
675 has to 1. 

Proposition 15. 

The diameter of the sun has to the diameter of the earth 
a ratio greater than that which 19 has to 3, but less than that 
which 43 has to 6. 

For let A be the centre of the sun, Β the centre of the earth, 
C the centre of the moon when the eclipse is total, so as to secure 
that A, B, C may be in a straight line. 

D d 2 



I 



404 



ON THE SIZES AND DISTANCES 



και ποιείτω τομά? εν μεν τω ήλίω τον ΔΚΖ κύκλον, εν δε τη γί) τον 

ΗΘΚ, εν δε τω σκιάσματι την Να περιφερειαν, εν δε τω κώνω 

ευθεία? τά? AM, ΖΜ, και επεζεύχθω ή Να, κα\ άπο του Α τ$ AM 

προς ορθά? ήχθω ή Ο ΑΠ. και 
5 €7Γ€ί ή Να τη? διαμέτρου του 

ήλιου ελάσσων εστίν η θ μύρο?, 

ή 0H άρα προ? την Να πολλω 

μείζονα λόγον έχει ή ον τά θ 

προ? α• και ή AM άρα προ? 
ίο την MP μείζονα λόγον έχει ή 

ον τα β προ? α. και άνα- 

στρεψαντι ή ΜΑ προ? ΑΡ 

ελάσσονα λόγον έχει ή ον τά 

θ προ? η. πάλιν επει ή ΑΒ 
15 τη? ΒΓ μείζων εστίν η ιη } 

πολλω άρα τη? Β Ρ μείζων 

εστίν η ιη' η ΑΒ αρα προ? την 

Β Ρ μείζονα λόγον έχει η ον 

τά ιη προ? α. άνάπαλιν άρα 
20 η ΒΡ προ? την ΒΑ ελάσσονα 

λόγον έχει ή ον α προ? ιη. 

και συνθίντι ή ΡΑ άρα προ? 

την ΑΒ ελάσσονα λόγον έχει 

η ον τά ιθ προ? τά ιη. εδείχθη 
25 δε και ή ΜΑ προ? την ΑΡ 

ελάσσονα λόγον έχουσα ή ον τά 

ό προ? τα η' εξει αρα οι ίσου 

ή ΜΑ προ? την ΑΒ ελάσσονα 

λόγον ή ον τά ροα προ? ρμδ, 
3ο και ον τά ιθ προ? ί<τ* τά 




Fig• 34• 



γάρ μέρη τοΐ? ώσαύτω? πολλαπλάσιοι? τον αύτον έχει λόγον 

6. θ'] εννατον W μέρος] η αρα διάμετρος του ηλίου μείζονα λόγον έχει προς την 

ΝζΙ ή ον τα θ προς α• κα\ add. W g. προς α] αλλ' ως η ΟΠ προς την Ν α, τουτέστιν, 

ως η ΑΟ προς την ΡΝ, όντως η AM προς την ΡΜ, δί' ομοιότητα τριγώνων add. W 
και] om. W ΙΟ. MP] PMW μείζονα] πολλω μείζονα W Ι φ η] τά η W 

15» 17 • ίΤ )] όκτωκαώεκαπλασίων W $1. ωσαύτως] ώσαυτωι Vat. 



OF THE SUN AND MOON 405 

Let a plane be carried through the axis, 
and let it cut the sun in the circle DEF, the earth in GHK, the 
shadow in the circumference NO, and the cone in the straight 
lines DM, FM. 

Let NO be joined, and from A let PAQ be drawn at right 
angles to AM. 

Then, since NO is less than i/c;th part of the diameter of the 
sun, [Prop. 13] 

therefore PQ has to NO a ratio much greater than that which 
9 has to 1. 

Therefore AM also has to MR a ratio greater than that 
which 9 has to 1 ; 

and, convertendo, MA has to AR a ratio less than that which 
9 has to 8. 

Again, since A Β is greater than 18 times BC, [Prop. 7] 

therefore it is much greater than 18 times BR ; 
therefore A Β has to BR a ratio greater than that which 18 
has to 1 ; 

therefore, inversely, BR has to Β A a ratio less than that which 
1 has to 1 8 ; 

therefore, componendo, RA has to A Β a, ratio less than that 
which 19 has to 18. 

But it was proved that MA also has to AR a ratio less than 
that which 9 has to 8 ; 

therefore, ex aequali, MA will have to A Β a ratio less than that 
which 171 has to 144, and therefore less than that which 19 has 
to 16 : for parts have the same ratio as the same multiples ot 
them: 



406 ON THE SIZES AND DISTANCES 

άναστρεψαντι άρα ή AM προς Β Μ μείζονα λόγον έχει ή ον τά ιθ 
προς τά γ. ως δε ή AM προς MB, όντως ή διάμετρος τον ΔΕ Ζ 
κνκλον προς την διάμετροι/ τον ΗΘΚ κνκλον ή άρα τον ήλίον 
διάμετρος προς την της γης διάμετρον μείζονα λόγον έχει η ον τα ιθ 

5 προς γ. 

Λέγω δη οτι ελάσσονα λόγον έχει (προς αντην) ή ον τα, μγ προς 
ς. επει γαρ ή ΒΓ προς την ΤΡ μείζονα λόγον έχει ή ον τα. χοε προς 
α, άναστρεψαντι άρα ή ΤΒ προς την Β Ρ ελάσσονα λόγον έχει ή ον 
τα, χοε προς τα χοδ. έχει δε και ή ΑΒ προς την Β Γ ελάσσονα 

ίο λόγον η ον τα κ προς α• εξει άρα δι 'ίσον η ΑΒ προς την Β Ρ 

α 

ελάσσονα λόγον η ον τα Μ γφ προς τά χοδ, τοντεστιν, η ον τα 

ς-ψν προς τα, τλζ• άνάπαλιν άρα καϊ σννβεντι ή ΡΑ προς την ΑΒ 

μείζονα λόγον έχει η ον τα, ζπζ προς ?ψν. και επεϊ ή ΝΈ προς 

α 

την ΟΠ μείζονα λόγον έχει ή (ον τα) ^)οθ προς Μρκε, άνάπαλιν 

α 

J 5 άρα ή ΟΠ προς Να ελάσσονα λόγον έχει η (ον τα) Μρκε προς 
~^οθ- ως δε ή ΟΠ προς Ν&, όντως ή AM προς MP- και ή AM 

α 

άρα προς MP ελάσσονα λόγον έχει ή (ον τα) Μρκε προς ~^ο0• 
άναστρεψαντι ή ΜΑ άρα προς την ΑΡ μείζονα λόγον έχει η ον τα, 

α 

Μρκε προς τά θρμς. έχει δε και ή ΡΑ προς ΑΒ μείζονα λόγον η 
20 ον τά ζπζ προς τά <τψν δι ΐσον άρα εξει ή ΜΑ προς την ΑΒ 

α 

μείζονα λόγον η ον ό περιεχόμενος αριθμός νπο των Μρκε και των 
ζπζ προς τον περιεχόμενον αριθμόν νπό τε των θρμς καϊ των 

ςψν, τοντεστιν, ό Μ εωοε προς Μ εφ. έχει δε και ό Μ εωοε 

,Γρογ 

προς Μ εφ μείζονα λόγον ή &ν τά μγ προς λζ• και ή ΜΑ άρα προς 
a5 την ΑΒ μείζονα λόγον έχει ή ον μγ προς λζ• άναστρεψαντι άρα ή 



I. AM] ΑΒ W 6. δη] δε W (προς αντην) addidi 8. Άρα] e£ei αρα W, 

α 

qui lacunam postl. 7 χ°* expleverat έχει] om. W χι. Μ,γφ] Μα.,γφ W 

α 

12. τ\ζ] τ\ν Vat. 13. £ψν] τα ,ς^ν W 14, 15, 17» *9, 21. M P Kf ] Μα.ρκ* W 

ΐ6. ~^>ο0] τα ~^οθ W \η. ~V0] τά ~~^οθ W 21-2. των ,ζπζ] τον ,ζπζ Vat. 

,fpoe ΛΡΟΥ 

23, 24. M.ecooe] M/pof και ,e&>oe W, bis Μ,εφ] Μ,ςρογ και ,(φ W, bis (haud recte) 



OF THE SUN AND MOON 407 

therefore, convertendo, AM has to Β Μ a ratio greater than that 
which 19 has to 3. 

But, as A Μ is to MB, so is the diameter of the circle DEF to 
the diameter of the circle GHK ; 

therefore the diameter of the sun has to the diameter of the earth 
a ratio greater than that which 19 has to 3. 

Again, I say that it has to it a ratio less than that which 43 
has to 6. 

For, since BC has to CR a ratio greater than that which 675 
has to 1, [Prop. 14] 

therefore, convertendo, CB has to BR a ratio less than that which 
675 has to 674. 

But A Β also has to BC a ratio less than that which 20 
has to 1 ; [Prop. 7] 

therefore, ex aequali, A Β will have to BR a ratio less than that 
which 13500 has to 674, that is, than that which 6750 has to 337 ; 
therefore, inversely and componendo, RA has to A Β a ratio 
greater than that which 7087 has to 6750. 

Now, since NO has to PQ a ratio greater than that which 979 
has to 1 01 25, [Prop. 13] 

therefore, inversely, PQ has to NO 2l ratio less than that which 
1 01 25 has to 979. 

And, as PQ is to NO, so is AM to MR ; 
therefore AM also has to MR a ratio less than that which 10125 
has to 979 ; 

therefore, convertendo, MA has to AR 3. ratio greater than that 
which 1 01 25 has to 9146. 

But RA also has to A Β a ratio greater than that which 7087 
has to 6750 ; 

therefore, ex aequali, MA will have to A Β a ratio greater than 
that which the number representing the product of 10125 and 7087 
has to the number representing the product of 9146 and 6750, 
that is, 71755875 to 61735500. 

But 71755875 has to 61735500 a ratio greater than that which 
43 has to 37 ,• 1 

therefore MA also has to A Β a ratio greater than that which 43 
has to 37; 

1 As to this approximation see p. 336 ad fin. 



4 o8 ON THE SIZES AND DISTANCES 

AM προ? τήν MB ελάσσονα λόγον έχει ή ον τα μγ προς q. ω? δε 
ή AM προ? την ΒΜ, όντως εστίν η διάμετρος τον ήλίον προ? την 
διάμετρον τη? γη?• ή άρα διάμετρο? τον ήλίον προ? την διάμετρον 
τή? γή? ελάσσονα λόγον έχει ή ον μγ προ? ς-, εδείχθη δε και 
5 μείζονα λόγον εχονσα ή ον τα ιθ προ? τα y. 



Ό ήλιο? προ? τήν γήν μείζονα λόγον έχει ή ον ς-ωνθ 

ζ 
προ? κζ, ελάσσονα δε ή ον Μ θφζ προ? σις, 

"Εστω γαρ ήλίον μεν διάμετρο? ή Α, γή? δε ή Β. άποδείκννται 
ίο δε οτι εστίν, ώ? ή τον ήλίον σφαίρα προ? την τή? γή? σφαϊραν, 
οντω? ό άπο τή? διάμετρον τον ήλίον κνβο? προ? τον άπο τή? 
διάμετρον τή? γή? κνβον, ωσπερ και επι τή? σελήνη?, ώστε επεί 
εστίν, ώ? ό άπο τή? Α κνβο? προ? τον άπο τή? Β κνβον, οντω? 
ό ήλιο? προ? τήν γήν, ό δε άπο τή? Α κύβο? προ? τον άπο τή? 

15 Β (κνβον) μείζονα λόγον έχει ή ον τα <?ωνθ προ? κζ } ελάσσονα δε 

ζ 

ή ον Μ θφζ προ? σις' και γαρ ή Α προ? τήν Β μείζονα λόγον έχει 

ή ον ιθ προ? γ, ελάσσονα δε ή ον μγ προ? ς** ώστε ό ήλιο? προ? 

τήν γήν μείζονα λόγον έχει ή ον ^ωνθ προ? κζ, ελάσσονα δε ή ον 

ζ 
Μ θφζ προ? σίς; 

30 ίζ . 

Ή διάμετρο? τή? γή? προ? τήν διάμετρον τή? σελήνη? 
εν μείζονι μεν λόγω εστίν ή ον (έχει) ρη προ? μγ, εν 
ελάσσονι δε ή ον ξ προ? ιθ. 

ι. την MB] MBW 2. η AM] AM W 

ζ 
6. ις ] IE Vat. 8. κζ] τα κζ W Μβφζ] μυριάδες ζ και βφζ W σις] ις Vat. 

et codd. Paris., excepto 2364 9. γης δε] γης W II. διαμέτρου] διαμέτρου 

της W 14. τον] την Vat. 15. (κύβον)] om. Vat. et codd. Paris. 16. Μβφζ] 

Μζ.βφζϊΝ σις] iS" Vat. 17. ώστε] apodosis hie desideratur ; exspectaveris 

ζ 
δια. ταύτα δη 6 ήλιος vel ό ήλιος αρα 19. Μβφζ] μυριάδες ζ κα\ βφζ W 

20. ι£] IT Vat. 22. ρη] τα. ρη W 






OF THE SUN AND MOON 409 

therefore, convertendo, AM has to MB a ratio less than that 
which 43 has to 6. 

But, as AM is to BM, so is the diameter of the sun to the 
diameter of the earth ; 

therefore the diameter of the sun has to the diameter of the earth 
a ratio less than that which 43 has to 6. 

And it was before proved that it has to it a ratio greater than 
that which 19 has to 3. 



Proposition 16. 

The sun has to the earth a ratio greater than that which 6859 
has to 2*], but less than that which 79507 has to 216. 

For let A be the diameter of the sun, Β that of the earth. 
Now it is proved that, as the sphere of the sun is to the sphere 
of the earth, so is the cube on the diameter of the sun to the cube 



Β 



Fig. 35• 

on the diameter of the earth, just as in the case of the moon [cf. 
Prop. 10]. 

Thus, since, as the cube on A is to the cube on B, so is the sun 
to the earth, 

while the cube on A has to the cube on^a ratio greater than that 
which 6859 has to 2j> but less than that which 79507 has to 216 : 
for A has to Β a ratio greater than that which 19 has to 3, but less 
than that which 43 has to 6 : [Prop. 15] 

it follows that the sun has to the earth a ratio greater than that 
which 6859 has to 2^ but less than that which 79507 has to 216. 



Proposition 17. 

The diameter of the earth is to the diameter of the moon 
in a ratio greater than that which 108 has to 43, but less than 
that which 60 has to 19. 



4io ON THE SIZES AND DISTANCES 

'Έστω γαρ ήλιου μεν διάμετρος ή Α, σελήνης δε ή Β, γή? δε 

ή Γ. και επει ή Α προ? την Τ ελάσσονα λογον έχει ή ον τα μγ 

προ? <?, άνάπαλιν άρα ή Τ προ? την Α μείζονα λογον έχει ή ον q 

προ? μγ. έχει δε και ή Α προ? την Β μείζονα λογον ή ον τά 

5 ιη προ? α• δι 'ίσου άρα ή Γ προ? την Β μείζονα λογον έχει ή ον 

Α 

α Β 

c β — 

7 

Fig. 36. 

τα ρη προ? τά μγ. πάλιν επει ή Α προ? την Γ μείζονα λογον έχει 
ή ον τά ιθ προ? τά γ, άνάπαλιν άρα ή Γ προ? την Α ελάσσονα 
λογον έχει ή ον τά γ προ? τά ιθ. έχει δε ή Α προ? την Β ελάσσονα 
λογον ή ον τά κ προ? α' δι ϊσου άρα ή Γ προ? την Β ελάσσονα λογον 
ίο έχει η ον ξ προ? ιό. 

ιη. 
Ή γή προ? την σελήνην εν μείζονι μεν λόγω εστίν 

pne ζ 

ή ον (β χ* ι) Μθψιβ προ? Μ^θφζ, εν ελάσσονι δε ή ον 

κα 

Μ <? προ? τωνθ. 
Ι5 "Έ,στω γαρ γή? μεν διάμετρο? ή Α, σελήνη? δε ή Β' ή Α άρα 
προ? την Β μείζονα λογον έχει ή ον τά ρη προ? τά μγ } ελάσσονα 
δε ή ον τά ζ προ? ιθ' και 6 άπο τή? Α άρα κύβο? προ? τον άπο τή? 

ρκε ζ 

Β κύβον μείζονα λογον έχει ή ον Μ θψιβ προ? Μ θφζ, ελάσσονα 

κα 

δε ή ον Μ q προ? ςωνθ. ώ? δε 6 άπο τή? Α κύβο? προ? τον άπο 
20 τή? Β κύβον, οΰτω? εστίν ή γή προ? την σελήνην ή γή άρα προ? 

p/ee ζ 

την σελήνην μείζονα μεν λογον έχει ή ον Μ θψιβ προ? Μ θφζ } 

κα 

ελάσσονα δε ή ον Μ q προ? ςωνθ. 

Ι σελήνη* δε ή Β] σελήνης ή Β W 5• «Ρ"] 7«Ρ Va *• 8 • ιθ ] & Vat • 

ή] και ή W 

p/ce 
II. irf] ΙΖ Vat. 13. («Χ") Μ,^ίβ] εχονσι μυριάδες ρκε κα\ βψιβ W 

ζ κα 

13, 18, 21. Μβφζ] Μζ.βφζνΐ 14, Ι9> 22. Μ,ς] Mica.fW 16. τά μγ] 

βΚ€ 

μ -yW ΐ8, 21. Μβ\\ηβ] Μρκε.βψιβ^ν 21. προς] προς μεν W 



OF THE SUN AND MOON 411 

For let A be the diameter of the sun, Β that of the moon, C that 
of the earth. 

Then, since A has to C a ratio less than that which 43 has to 6, 

[Prop. 15] 
therefore, inversely, C has to A a ratio greater than that which 
6 has to 43. 

But A also has to Β a ratio greater than that which 18 has to 1 ; 

[Prop.9] 
therefore, ex aequali, C has to Β a ratio greater than that which 
108 has to 43. 

Again, since A has to C a ratio greater than that which 19 
has to 3, [Prop. 15] 

therefore, inversely, C has to A a ratio less than that which 
3 has to 19. 

But A also has to Β a ratio less than that which 20 has to 1 ; 

[Prop. 9] 
therefore, ex aequali, C has to Β a ratio less than that which 60 
has to 19. 

Proposition 18. 

The earth is to the moon in a ratio greater than that which 
1 25971 2 has to 79507, but less than that which 216000 has to 

6859. 

For let A be the diameter of the earth, Β that of the moon ; 
therefore A has to Β a ratio greater than that which 108 has to 
43, but less than that which 60 has to 19. [Prop. 17] 

Therefore also the cube on A has to the cube on Β a ratio 



Β 



Fig. 37- 

greater than that which 12597 12 has to 79507, but less than 
that which 216000 has to 6859. 

But, as the cube on A is to the cube on B y so is the earth to 
the moon ; 

therefore the earth has to the moon a ratio greater than that which 
1259712 has to 79507, but less than that which 216000 has to 6859. 



412 ON THE SIZES AND DISTANCES 



COMMENTS OF PAPPUS. 1 

* In his book on sizes and distances Aristarchus lays down these 
six hypotheses : 

ι . That the moon receives light from the sun. 

2. That the earth is in the relation of a point and centre to 
the sphere in which the moon moves. 2 

3. That, when the moon appears to us halved, the great 
circle which divides the dark and the bright portions of the moon 
is in the direction of our eye. 

4. That, when the moon appears to us halved, its distance 
from the sun is then less than a quadrant by one-thirtieth of a 
quadrant. 3 

5. That the breadth of the (earth s) shadow is (that) of two 
moons. 

6. That the moon subtends one fifteenth part of a sign of the 
zodiac. 

Now the first, third, and fourth of these hypotheses practically 
agree with the assumptions of Hipparchus and Ptolemy. For the 
moon is illuminated by the sun at all times except during an 
eclipse, when it becomes devoid of light through passing into the 
shadow which results from the interception of the suns light by 
the earth, and which is conical in form ; next the (circle) dividing 
the milk-white portion which owes its colour to the sun shining 
upon it and the portion which has the ashen colour natural to the 
moon itself is indistinguishable from a great circle (in the moon) 
when its positions in relation to the sun cause it to appear halved, 
at which times (a distance of) very nearly a quadrant on the circle 
of the zodiac is observed (to separate them) ; and the said dividing 
circle is in the direction of our eye, for this plane of the circle 
if produced will in fact pass through our eye in whatever position 
the moon is when for the first or second time it appears halved. 

1 Pappus, vi, pp. 554. 6-560. 10 (Hultsch). 

2 Literally, ' the sphere of the moon.' 

3 Hultsch brackets, as an obvious interpolation, words added here in the 
Greek text ' instead of (saying) that its distance is 87 : for this is less than 
a quadrant, or 90 , by 3 , which is i/30th of 90° '. 



OF THE SUN AND MOON 413 

But, as regards the remaining hypotheses, the aforesaid mathe- 
maticians have taken a different view. For according to them the 
earth has the relation of a point and centre, not to the sphere in 
which the moon moves, but to the sphere of the fixed stars, the 
breadth of the (earth's) shadow is not (that) of two moons, nor does 
the moon's diameter subtend * one fifteenth part of a sign of the 
zodiac, that is, 2 . According to Hipparchus, on the one hand, the 
circle described by the moon is measured 650 times by the diameter 
of the moon, while the (earth's) shadow is measured by it 2 \ times at 
its mean distance in the conjunctions ; in Ptolemy's view, on the 
other hand, the moon's diameter subtends, when the moon is at its 
greatest distance, a circumference of o°3i / 2o // , and when at its 
least distance, of o° 35' 20", while the diameter of the circular 
section of the shadow is, when the moon is at its greatest distance, 
o° 40' 40", and when the moon is at its least distance, o° 46'. 

Hence it is that the authors named have come to different 
conclusions as regards the ratios both of the distances and of the 
sizes of the sun and moon. 

Now Aristarchus, after stating the aforesaid hypotheses, proceeds 
in a passage which I will quote word for word. 2 

11 We are now in a position to prove that the distance of the sun 
from the earth is greater than 18 times, but less than 20 times, the 
distance of the moon, and the diameter of the sun also has the same 
ratio to the diameter of the moon : this follows from the hypothesis 
about the halved (moon). Again, we can prove that the diameter 
of the sun is to the diameter of the earth in a greater ratio than 
that which 19 has to 3, but in a less ratio than that which 43 has 
to 6 : this follows from the ratio thus discovered as regards the 
distances, from the hypothesis about the shadow and from the 
hypothesis that the moon subtends one fifteenth part of a sign 
of the zodiac." 

He says " We are in a position to prove that the distances", &c, 

1 Hultsch brackets, as an interpolation, some clumsy words in the Greek text, 
the object of which is to qualify ' diameter ' and make it mean the diameter of 
the moon when it is 'at the same mean distance' ; there are no such words in 
Aristarchus. 

2 Gk. λέγων κατά λεξιν ούτως. Although Pappus professes to quote Aristarchus 
word for word, he shows some slight variations from the text of Aristarchus 
as we have it ; where the changes are for the worse, however, they may be due 
to copyists rather than to Pappus himself. 



414 



COMMENTS OF PAPPUS 



implying that he will prove the properties after giving such pre- 
liminary lemmas as are of use for the proofs of them. As the result 
of the whole investigation he concludes that 

(i) the sun has to the earth a greater ratio than that which 
6859 has to 27, but a less ratio than that which 79507 has to 216 ; 

(2) the diameter of the earth is to the diameter of the moon in 
a greater ratio than that which 108 has to 43, but in a less (ratio) 
than that which 60 has to 19 ; and 

(3) the earth is to the moon in a greater ratio than that which 
1 2597 1 2 has to 79507, but in a less (ratio) than that which 
216000 has to 6859. 

But Ptolemy proved in the fifth book of his Syntaxis J that, if the 
radius of the earth is taken as the unit, the greatest distance of 
the moon at the conjunctions is 64 J§ of such units, the greatest 
distance of the sun 12 10, the radius of the moon ^J §§ 2 , the radius 
of the sun 5§g. Consequently, if the diameter of the moon is 
taken as the unit, the earth's diameter is 3§ of such units, and the 
sun's diameter i8f. That is to say, the diameter of the earth is 
3§ times the diameter of the moon, while the diameter of the sun 
is 18 § times the diameter of the moon and 5I times the diameter ot 
the earth. 

From these figures the ratios between the solid contents are 
manifest, since the cube on 1 is 1 unit, the cube on 3•! is very 
nearly 39J of the same units, and the cube on i8f very nearly 
6644I, whence we infer that, if the solid magnitude of the 
moon is taken as a unit, that of the earth contains 39J and that 
of the, sun 6 644 \ of such units ; therefore the solid magnitude of 
the sun is very nearly 170 times greater than that of the earth.' 

1 Ptolemy, Syntaxis, v, 14-16, vol. i, pp. 416-427, Heib. 



INDEX 



Achilles (not Tatius) 5, 112, 116, 174. 

Adam, Dr. lion., 138-9, 149-52, 153, 
156, 171-2. 

Adrastus (in Theon of Smyrna) 6, 112, 
199, 256, 257-8, 268, 270, 342. 

Aeschylus, pupil of Hippocrates of 
Chios, 243-4 

Aetius, 77-epl άρ^σκόντων (De ftlacitis), 4 
and passim. 

Agesilaus 192. 

Alcmaeon of Croton 49-50, 103, 107. 

Alexander Aphrodisiensis 98, III, 129, 
176, 187, 275, 282. 

Alexander of Ephesus (or Miletus) 
1 12-14. 

Alexander Polyhistor 2, 65 n. 

Anatolius 115. 

Anaxagoras 18, 30,40, 44, 48, 50, 78-85, 
143 : cosmogony through motion of 
vortex and * hurlings-off ' (centri- 
fugal force) 81-2 : other worlds than 
ours 85 : stars are stones on fire 81-2, 
kindled by rapid motion 81 : stars 
first revolved horizontally, then axis 
tilted 82-3, 91, 92 : stars more dis- 
tant than sun and moon 82 : remark- 
able theory of Milky Way 83-5 : 
order of planets 85, 128 : sun a red- 
hot mass or stone on fire 82, larger 
than the Peloponnese ibid. : earth 
(and probably other bodies) flat 81, 
91-2, 144, 146 : earth rests on air 8^ f 
144, 238 : discovered that moon is lit 
up by sun 19, 76, 77, 78-9, 9i> 158 : 
on substance and light of moon 82 : 
other dark bodies besides earth and 
moon cause eclipses 79-80 : on 
comets 125, 243: story of meteoric 
stone 246. 

Anaximander 4, 24-39, 55, 114 : work 
About Nature 24 : cosmogony 25-7, 
28-30 : innumerable worlds 29-30 : 
sun moon and stars hoops or wheels 
with vents 27,28,31: position of hoops 
31, 33-6 : sizes of hoops of sun and 



moon, and of sun and moon them- 
selves (first speculation on sizes and 
distances) 27, 28, 32, 37, 38, 114: 
stars nearer than sun and moon 28 : 
earth a short cylinder 25, and sus- 
pended freely 24, 25, 64, 145 : drew 
first map of inhabited earth 38, 39, 
124, 145 : said to have introduced 
gnomon 38 and constructed sphere 
38, 319 : evolutionary theory 39 n. 

Anaximenes 19, 27 n. y 33 n., 40-5 : 
on nature of heavenly bodies 40 : 
stars fixed like nails on crystal 
sphere 40, 45, except (presumably) 
planets 42, 43, 50 : stars more dis- 
tant than sun 43 : stars revolve 
laterally round, not under, earth 41- 
3, 83 : earth flat and rides on air 40, 
83, 144, 238: earthy bodies moving 
about among stars 43, 44, significant 
with reference to eclipses 44-5 (cf. 
Anaxagoras) : universe breathes 45. 

Αναφορικό? (De ascensionibus) of 
Hypsicles 317, 321, 325. 

Antipodes 65. 

Antisthenes, author of 8ιαδοχαί, 2. 

Apelt, E. F. 104, 194. 

Apollodorus of Athens, grammarian, 5. 

Apollodorus of Corcyra 307. 

Apollonius of Perga 193, 299: had 
theory of epicycles in all its generality 
266-7, 274, but thought eccentrics 
only applicable to superior planets 
266-8, 274 : possibly originated 
hypothesis of Tycho Brahe 269, 274. 

Aquinas, Thomas, 177. 

Arabic versions of Aristarchus 320-1: 
MSS. of, 321, 323. 

Aratus 23, 112, 222 n. 

Archedemus 187. 

Archelaus 124, 146. 

Archer-Hind 145 n., 167-8, 177, 180. 

Archimedes 23, 221, 299, 366 n. : on 
Democritus's discovery of volume of 
pyramid and cone 121 : on Aristar- 



1110 



4i6 



INDEX 



chus's heliocentric hypothesis 302-4, 
on other hypotheses of Aristarchus 
302, 308-9 : on apparent diameter 
of sun (a) as discovered by Aristar- 
chus 311, 353 n., (b) as measured by 
himself 312, 348 : on estimates of 
relative size of sun and moon 332, 
337 : on estimates of earth's circum- 
ference 147, 337-8, 347 : allowed for 
parallax in case of sun 348. 
Archytas, as geometer, 190, 191, 192, 
299 : on mathematics 319-20. 

αριθμητική 135• 

Aristarchus of Samos 282 : ' the mathe- 
matician,' pupil of Strato 299: on 
vision, light and colours 300 : cos- 
mology ibid. : inventor of a sun-dial 
(σκάφη) 312 : originator of heliocen- 
tric hypothesis 301-6, attacked for 
it by Cleanthes 304 : earth negligible 
in size compared with universe 302, 
308-9, and (in extant treatise) even 
with orbit of moon 309, 353, 412 : 
moon a satellite of earth 310.: found 
angular diameter of sun to be \° but 
in treatise assumes value 2 , 23, 311- 
12, 353, 412 : ' Great Year' of 2434 
years and solar year of 365J y^s 
days 314-16, explanation of these 
figures ibid. : treatise on sizes and 
distances 317, Arabic translations 
320-1, editions 321-4, MSS. 325-7, 
style 328, trigonometrical bearing of 
geometry in, 328, contains no trace 
of heliocentric hypothesis 310, 
assumptions in, as to apparent dia- 
meter of sun 353, 412, as to equality 
of apparent diameters of sun and 
moon 383, 412, as to diameter of 
earth's shadow (=2 diameters of 
moon) 329 sq., 337, 353, 412, as to 
angle subtended at centre of sun by 
line joining centres of earth and 
moon at half-moon 329, 337, 353, 
412: trigonometrical equivalent of 
propositions 333-6 : summary of 
main results 338, 350,413-14: text 
and translation of treatise 351-41 1. 

Aristarchus of Samothrace 10 n. 

Aristotherus 222. 

Aristotle 1, 9, 12 n., 18, 30, 32, 41, 
47, 48, 49, 50, 61, 83 : criticisms 
of views of predecessors regarding 
the earth, Thales 18 n., 19 n., 238, 
Anaximander 24-5, 239, Anaxi- 
menes, Anaxagoras and Democritus 



40, 238, Xenophanes 54, Empedocles 
88, 239 : on Heraclitus's sun 60 n. : 
on Anaxagoras's and Democritus's 
theory of Milky Way 83-5, 247 : 
on Empedocles' theory that light 
travels 93 : on Pythagorean system 
of ten ' bodies ' revolving round 
central fire 95-6, 98, 99, 100, 103, 
237 (* outermost revolution ' a 'physi- 
cal body ' 233) : reference to others 
who displaced earth from centre 
186-7, 273 : on supposed rotation 
of earth in Timaeus 174, 176-8, 240 : 
on Eudoxus's system of concentric 
spheres 193—5, 196, 197, 198, and 
Callippus's improvements 212 : A.'s 
own modification of system of con- 
centric spheres in mechanical sense 
by means of reading spheres (αι/βλίτ- 
τονσαι) 217-21, 225 : supposed doubts 
as to concentric system 222, 223, 
261 : A.'s own views, on motion and 
firimum movens 225-7, kinds of 
motion appropriate to elements 227, 
aether 227-8 : universe one 228-9, 
finite 229, spherical 229-30 : eternal 
movements other than that of whole 
universe 230-1 : stars spherical 233 : 
stars do not move of themselves but 
are fixed on spheres which move 
106 n., 233-4 : on question whether 
moving heavenly bodies produce 
sound 105-6, 108 : heavenly bodies 
do not rotate or roll 234-5 (A• does 
not deny incidental rotation of moon 
235) : on planets' obliquities 155 n. : 
earth a sphere, reasons for this view 
2 35~7 : attempt to prove earth at 
rest in centre 237-41 : on size of 
earth 147, 236 : on ' right ' and ' left ' 
in universe 161, 231-2 : on ' up ' and 
' down ' 237-8 n., 239 : on four ele- 
ments, their 'causes' and inter- 
changes 241-2: on shooting stars 
and meteors 219, 242-3 : on comets 
219, 243-7 : on Milky Way 219, 
247-8 : on the tides 306. 

Aristoxenus 5. 

Ars Eudoxi, or Didascalia caelestis of 
Leptines 112, 200 #., 208, 293. 

Assyrian predictions of eclipses, 16, 17. 

Astronomy, ' Treasury ' of (άστρονο- 
povpevos τόπος), or ' Little Astro- 
nomy' (ρικρος άστρονυρούμβνος) $17~ 
2o: Arabic translations of, 320-1. 

Athenaeus 11. 



INDEX 



417 



Autolycus of Pitane 221, 261 : works 
On the moving sphere and On risings 
and settings 192, 317-18, 321, 325. 

Babylonians, used gnomon and polos 
38 : unacquainted with sphericity of 
earth 48 n. : unacquainted with pre- 
cession in third cent. B.C., 105 : esti- 
mate of apparent diameter of sun 
22, 311 : reference in Aristotle to, 
220 : see also Chaldaeans. 

Bear, Great, non-setting in Homer's 
time, but not so now in Mediterra- 
nean 8-9 n. : Eudoxus and Proclus 
on position of, ibid. 

Bear, Little, observed by Thales 23 : 
Phoenician navigators set their 
course by, 23. 

Berger 39 η , 48 λ., 55 η., 56-8, 154-5 »•, 

337 «•, 344, 345 «• 

Bergk 277, 279-80, 302 »., 303-4 η., 307. 

Bernard, Edward, 323. 

Berosus 16 «., 314. 

Berry, A. 343 n. 

Bilfinger 22 n. f 193 n. 

Biographies, by Satyrus, Heraclides 
Lembus, Hermippus, Laertius Dio- 
genes 2-3. 

Bjornbo, A. A. 318 n. 

Boeckh 101, 105, 117, 118, 150- 1,1 52 n., 
153, 161-2, 163, 175, 177, 178, 179, 
180, 183, 186, 231 n^ 250, 258»., 
277, 290, 291, 293, 295. 

Boll, F. 14 n. f 16 «., 76, 80 n. t 343 n. 

Bosanquet 139-40. 

Brandis 22 n. 

Burnet (Early Greek Philosophy) 6, 
25 n^ 29, 30 n.f 46 n.f 47 «., 50, 53, 
62, 67 η.^ 69 n.f 85 n.f 92 «., 100 n. y 
103, 105 n.f 107, 116, 117, 121, 173. 

Callippus 129, 193, 197, 200, 293: im- 
provements on Eudoxus's theory of 
concentric spheres 212-16, 217, 219, 
220, 221, 225, 261, 278 : on length 
of seasons 215-16 : cycle of 76 years 
((κκακβδομηκονταίτηρίί) 295-6. 

Canopus, the star, 192 «., 345. 

Cantor, M. 16 n. 

Cassini 343. 

Censorinus 107 n. } 113, 114, 129, 132, 
284, 286, 291, 292, 293, 314, 315. 

Chalcidius 164, 167, 270, 342: on 
Heraclides' theory of Venus and 
Mercury 256-8. 

Chaldaeans : predictions of eclipses 



by means of periods of 223 luna- 
tions 16 (cf. 314, 315) : eclipses 
observed in Babylon in 721-0 B.C., 
16 : Chaldaean order of planets 
258, 259. 

Cicero 4, ι$η., 29, in, 167, 178, 188, 
252, 253, 256, 258, 259. 

Cleanthes, the Stoic, 74 n. : attacked 
Aristarchus for his heliocentric 
hypothesis 304. 

Cleomedes 22, 23, 8o#., 155 n. y 223-4, 
310, 313, 339, 342 n.f 344-7. 

Cleostratus, Astrology 23 : connected 
with octaeteris 291. 

Comets, views on : Anaxagoras and 
Democritus 125, 243, 245, certain 
Pythagoreans 243-4, Hippocrates of 
Chios and Aeschylus 243-4, Aris- 
totle 243-7, Heraclides 254, Seneca 
247 n. : particular comets mentioned 
by Aristotle 244-6. 

Commandinus, translation of Aristar- 
chus 321, 323. 

Continued fractions, application of, 
probably as early as Aristarchus 
336. 

Cook Wilson 156. 

Counter-earth of Pythagorean system 
96-8 : invented to explain eclipses 
99-100, 119, abandoned early 249: 
identified with moon 250. 

Copernicus : allusions of, to Philolaus, 
Heraclides, Ecphantus, and Hicetas 
301 : was aware of Aristarchus's 
hypothesis, ibid. : on distance of sun 

343• 

Crates of Mallos 305, 306-7. 

Cycles (of years) : reputed trieteris and 
tetraeteris or pentaetens (q.v.) 286 : 
octaeteris 287-92 : 16-years' {ίκκαι- 
δαίαίτηρίς) and i6o-years' periods 
292-3 : Meton's cycle of 19 years (eV- 
νςακαιδΐκαπηρίς) 293-5 : Callippus's 
cycle (76 years) 295-6 : Hipparchus's 
cycle (304 years) 296-7. 

Damastes, of Sigeum, 124. 

Day, in Greece, was from one sunset 
to next 284. 

Delambre 17 n. 

Democritus 26, 40, 50, 64 : discovered 
volumes of pyramid and cone, and 
was on track of infinitesimals 12 1-2 : 
in astron. generally followed Anaxa- 
goras 123 : sun and stars red-hot 
stones, moon has light from sun 



1410 



Ε e 



4i 8 



INDEX 



123-4 : on comets 125, 243, 245 : on 
Milky Way 83, 124-5 : infinite num- 
ber of worlds 125 : our sun and moon 
originally nuclei for other worlds 
127-8 : on planets and their order 
128 : on relative speeds of sun, moon, 
and planets 128-9: earth a disc 
hollowed out in middle 124, and 
elongated (προμήκης) 84, 1 24 : earth 
in equilibrium without support (cf. 
Anaximander and Parmenides) 64, 
124, or rides on air 144, 238 : ' Great 
Year' 129: geographical and nauti- 
cal survey of inhabited earth 124, 
145 : Gomperz's rhapsodical estimate 
of Democritus's astronomy 125-7. 

De Morgan 139. 

Dercyllides (in Theon of Smyrna) 6, 

131, 183, 304, 307• 
διαδοχαί, * Successions,' authors of, 2. 

Diakosmos, Great, of Leucippus 122. 
Dicaearchus 306, 337, 339, 347. 
Diels : Doxographi Graeci 2-5, and 

Fragmente der Vorsokratiker 5, 

passim : 27 «., 37-8, 52, 67, 70-4, 

76, 79 n. y 92 n., 97 n., 122 «., 277, 

279, 281, 295 n. 
Diodorus, of Alexandria, Astronomy 5. 
Diodorus Siculus 17, 131, 293, 294, 

295. 
Diogenes of Apollonia 2, 246 n. 
Diogenes of Babylon 107. 
Diogenes Laertius 2-3, and quoted 

Passim. 
Dioscuri 55. 
Dositheus 291. 
Doxographi Graeci, Diels' account of, 

2-5 : genealogical table 3. 
Dreyer, J. L. E. 33 n., 82 «., 90, 

177 n. t 195 n., 220 «., 235 «., 240, 

340 n. 



Earth : sphericity of, first maintained 
by Pythagoras and Parmenides 21, 
48, 49, 64, unknown to Babylonians 
and Egyptians 48 n. : rotation of, 
affirmed by Heraclides 251, 254-5 
and Aristarchus 304 (discovery ques- 
tionably attributed to Hicetas 187-9, 
and to Ecphantus 251-2, 282),admit- 
ted as possibility by Seneca 307-8 : 
size of, earliest estimates 147, 236, 
Eratosthenes' measurement, 114, 
I 47> 339-40> Posidonius's measure- 
ments 345-7 : size relatively to sphere 



of universe, &c. 308-10 : as ' instru- 
ment of time ' 250. 

Earth's shadow, diameter of, at dis- 
tance of moon, as estimated by 
Aristarchus, Hipparchus, and Pto- 
lemy 329, 337, 353,412,413. 

Eccentrics, movable, 263 sq. : equiva- 
lent to epicycles 264-6 : with Apollo- 
nius application limited to superior 
planets 266-7, 274 : theory gene- 
ralized by Hipparchus's time 267- 
8 : not invented by Pythagoreans 
270-4. 

Ecphantus, credited with theory of 
earth's rotation 251, 282: probably 
a personage in one of Heraclides' 
dialogues 251-2. 

Eclipses : supposed prediction of solar 
eclipse by Thales 13-18, by Helicon 
13-14, 193 : date of Thales' eclipse 
15-16: period of recurrence (223 
lunations) discovered by Chaldaeans 
16, and known to Thales ibid., per- 
haps to Egyptians 17: cuneiform 
inscription on observations of pre- 
dicted eclipses 16-17 : causes known 
to Pythagoreans 119:' paradoxical 
case,' and true explanation given by 
Cleomedes 80 n. : Sosigenes on 
annular eclipses of sun 222-4, 3 J 3> 

383»• 

Ecliptic (or zodiac circle) : discovery 
of obliquity attributed to Oenopides 
21, 1 30- 1, 319, possibly learnt from 
Egyptians 131, 319: estimate of 
obliquity 24 , discovered before 
Euclid's time 131 «., and possibly 
still used by Eratosthenes and Hip- 
parchus ibid. : later estimate, half of 
|£ of 360 , attributed to Eratos- 
thenes but probably Ptolemy's ibid. : 
another value incidentally given by 
Pappus 132 n. 

Egyptians : predicted eclipses 17 : 
possibly discovered obliquity of 
ecliptic 131, 319 : unaware of spheri- 
city of earth 48 n. : on order of 
planets 258 : Heraclides' theory 
regarding Venus and Mercury 
wrongly attributed to, 259 : had year 
of 365 days and month of 30 days 
21 : drew maps 38. 

€κκαιδ€κα(τηρίς (i6 years' period) 292-3. 

€κκακβδομηκοι>τα€τηρίε (76 years' cycle 
of Callippus) 295-6. 

Ell (πήχυς), Babylonian : use of, as 



INDEX 



419 



astronomical measure of angles 

(=2°) 23 Λ. 

Empedocles : date, &c. 86 : poems 87 : 
stars fixed on crystal sphere, which 
is egg-shaped 87 : explanation of 
night and day by dark and light 
hemispheres revolving 87-8: earth 
kept in place by swift revolution of 
heaven 88, 144, 239 : sun explained 
as reflection of fire in universe 89- 
90 : on tropic circles limiting motions 
of sun 87, 91 : substance of moon 
and stars 91 : moon lit up by sun 
ibid. : earth flat (probably) 91 : axis 
of earth originally perpendicular to 
surface 91-2, subsequently became 
tilted 92 : fires in centre of earth 92 : 
sun most distant of heavenly bodies 
87, and twice as distant as moon 92 : 
gave true explanation of eclipses 92 : 
greatest scientific achievement was 
theory that light travels 92-3. 

€Ρν€ακαιδ€κα(τηρίς (ig years' period, 
Meton's cycle) 293-5. 

Epicurus, on size of sun 61, 253 : on 
Heraclides 253. 

Epicycles : equivalence of, to movable 
eccentrics 264-6 : theory understood 
in all its generality (including epi- 
cycles with ideal points as centres) 
by Apollonius 266-7 •' not invented 
by Pythagoreans 270-4 : wrongly 
imported into explanations of Plato's 
and Heraclides' systems 166-7, 
256-8, 260. 

Efti?iomis, astronomy of, 184-5. 

Eratosthenes 112, 299, 345, 348: on 
measure of obliquity of ecliptic 131 — 
2 n. : measurement of circumference 
of earth 114, 147, 339-40: on dis- 
tances of sun and moon from earth 
340-1. 

Euclid : works included in i Little 
Astronomy', Phaenomena 309-10, 
317-19, Optics 317, 320. 

Euctemon : on length of seasons 200, 
213, 215: associated with Meton's 
cycle 293, 296. 

Eudemus, History of Astronomy, 6, 
14, 19, 20, 21, 24 «., 37, 131, 140, 
213, 272. 

Eudorus 5. 

Eudoxus 131 n., 141, 200, 278, 331 : par- 
ticulars of life 190-2: pupil of Archy- 
tas 190, 192: discoverer of general 
theory of proportion expounded in 



Eucl. V, and of method of exhaus- 
tion 191 : wrote Mirror (Ζνοπτρον) 
192, 199, Phaenomena 192, On 
Speeds 193, 320 n., and probably a 
work on Sfihaeric 192, 318 : invented 
arachne 193 : varied, if he did not 
lnventiOciaeteris (8-years' cycle) 291, 
293 : views on position of north pole 
8-9 n., on ratio of sizes of sun and 
moon (9:1) in, 332, 337, possible 
explanation of latter ratio 111-12 : 
theory of concentric spheres, de- 
scribed by Aristotle and Simplicius 
193-4,196-202, Schiaparelli's restora- 
tion of system 202-1 1 : hippopede 
described by planets 202, proof of 
shape 204-5 n. : supposed sun to 
deviate in latitude 198-9: ignored 
differences in length of seasons 200 : 
estimates of zodiacal and synodic 
periods of planets 208 : Ars Eudoxi 
112, 200 n., 208, 293 : perhaps 
originated method of Aristarchus's 
treatise on sizes and distances 331-2. 

Eusebius 4. 

Exeligmus 314-15. 

Fortia d'Urban, Comte de : Greek text 
and Latin translation of Aristarchus's 
treatise 323-4, 326 : French transla- 
tion of same 324 : MSS. used by, 
326-7 : 336. 

Fiilleborn 67. 

Galilei 126. 

Gallenmiiller 8n. 

Geminus 268, 269, 270, 272, 284, 286, 
287-8, 289, 292, 293, 294, 295, 296, 
310, 314-15 : famous passage quoted 
from, by Simplicius through Alex- 
ander 275-6. 

Gilbert, Otto 73, 74. 

Ginzel, F. K. 16 n., 132 n., 284 sq. 

Gnomon, introduced into Greece from 
Babylon 21, 38. 

Gomperz, Griechische Denker, 1 n., 6, 
18 n., 38 n., 46 n., 48 n., 49 n., 52 n., 
53/z., 8i«., 83, 84, 105, 125-7, 249 n., 
279, 281, 301 n. 

* Great Year,' of Heraclitus 61, Philo- 
laus 102, Democritus 129, Oeno- 
pides 102, 132-3, Plato 171-3, Aris- 
tarchus 314-16 : see also Cycles. 

Greenhill, Sir G. 8-9 n. 

Gruppe 175, 180, 182, 183, 186. 

Giinther, Siegmund 3 n % 



Eea 



42θ 



INDEX 



Hanno 249. 

Harmony of spheres, in Pythagorean 
system 98, 105-15, not mentioned by 
Philolaus 108 : possibly Pythagoras 
distinguished only three notes 107, 
but first form of harmony was of seven 
notes (heptachord) 107 : complete Py- 
thagorean system implied eight notes 
(cf. Plato) 108-11 : how heavenly 
bodies corresponded to eight notes 
(Plato, Nicomachus, Cicero, Boe- 
thius) ibid. : other lengths of scale, 
(a) octave of nine notes (Hypsicles) 
1 12-13, (b) scales of eight intervals 
adding up to 6, 6\ or 7 tones (Cen- 
sorinus, Pliny, Martianus Capella) 
ii3-i4,(£)otherdistributions, making 
up more than one octave (Plutarch, 
Anatolius, Macrobius) 115 : difficulty 
in deducing definite ratios of dis- 
tances III, but Pythagoras said to 
have made one tone (= 126,000 
stades) separate moon and earth 
114-15. 

Harpalus 291-2. 

Hearth of universe 97-9, 142, 304. 

Hecataeus 38, 39, 124. 

Heeren 43 n. 

Heiberg, J. L. 121 n., 303, 317 »., 
325 n. 

Heraclides Lembus, epitome of bio- 
graphies, 2, 5. 

Heraclides of Pontus 141, 167 : pupil 
of Plato 252 : particulars of life 252- 
3 : characteristics of dialogues 253 : 
Hicetas and Ecphantus probably 
personages in dialogues 188-9, 
251-2: views of H. on universe 
251, 252, 254 : universe infinite 254 : 
each star a universe ibid. : on moon, 
comets, and meteors ibid. : on the 
tides 306 : affirmed rotation of earth 
about its axis 251, 254-5, 282 : dis- 
covered that Mercury and Venus 
revolve round sun 25 5-60 : may pos- 
sibly have extended this theory to 
other planets and so invented system 
of Tycho Brahe 260-75, but evidence 
does not confirm this 269-75 : con- 
tention of Schiaparelli that H. was 
first enunciator of heliocentric hypo- 
thesis 275-9, but words in passage 
of Geminus relied on are clearly 
interpolated 280-2, so that Schia- 
parelli's argument fails 283. 

Heraclitus 50, 66 : on Xenophanes and 



Pythagoras 52 : crude astronomy 
59-61 : sun, moon, and stars are 
bowls collecting bright exhalations 
59-60 : eclipses due to turning of 
bowls upwards 44, 60 : new sun 
every day 60 : sun a foot in diameter, 
really as well as apparently 61 : 
'Great Year' 61. 

Hermippus the ' Callimachean ', writer 
of biographies, 2. 

Herodian of Alexandria 10 n. 

Herodotus 15, 21 «., 38, 286. 

Herz, Norbert, 203-5 n. 

Hesiod : astronomy in, 7-1 1, 12 : con- 
stellations mentioned by, 10 : times 
and seasons determined by risings 
and settings of stars 10-11 : men- 
tions solstices but not equinoxes 11, 
20 : supposed author of poem 
'Astronomy' 11, 23. 

Hicetas, of Syracuse, credited with 
discovery of rotation of earth about 
axis 187-8, or alternatively with 
Pythagorean system attributed to 
Philolaus ibid. : probably appeared 
as one of interlocutors in a dialogue 
of Heraclides 188-9. 

Hilprecht 105 n. 

Hipparchus : expressed angles in terms 
of ell 23 n. : observation of gnomon's 
shadow at Byzantium at summer 
solstice 131 n. : on position of north 
pole 8 n. : on latitude of star Cano- 
pus 192 n. : discovery of precession 
101, 172-3, 200, and estimate of its 
rate 172-3: denied that sun's orbit 
was inclined to ecliptic 199 : distin- 
guished two anomalies, solar and 
zodiacal 267-8 : on solar parallax 
13, 341 : upheld geocentric view 
308 : on epicycles and eccentrics 
267-8 : on apparent diameter 
of moon 313, 413 : on diameter of 
earth's shadow 329, 337, 413 : esti- 
mates of distances of sun and moon, 
and of sizes relatively to earth 342-3, 
350: view as to circumference of 
earth 114, 343-4, and measure of 
obliquity of ecliptic 132 n. : 258, 
278, 344, 412. 

Hippasus, the Pythagorean, 47. 

Hippocrates of Chios, on comets 

243-4. 
Hippopede of Eudoxus 202 sq. 
Homer : astronomy in, 7-9 : stars and 

constellations mentioned by, 8-9: 



INDEX 



421 



Great Bear non-setting in his time 
8-9 and note : τροπαί ψλίοιο 9~ιο, 
10 η. : divisions of day and night 9. 

Hultsch 22 η., 23, 65, 115, \6y η., iji, 
258 n., 259, 272, 303, 311 »., 313 n., 
338, 339^•» 34i> 342, 343, 344 n. f 
345; 346, 347, 349-50, 412 »., 413 »• 

Hypsicles : Αναφορικός 317, 321, 325 : 
wrote on harmony of spheres 112-13. 

Iamblichus 271. 

Ideler 10 n., 11 n., 193-4, 197, 247 «., 

290. 
ΙΚλομίνψ (in Timaeus 40 b) 175-7. 
Ishaq b. Hunain 321. 

Josephus 2. 

Kant 81. 

Kugler, Pater, 105 n. 

Laplace 31, 81. 

Lepsius 101. 

Leptines' Didascalia caelestis — Ars 
Eudoxi 112, 200, 208, 293. 

Leucippus 26, 12 1-3, 128 : cosmogony 
122 : earth like tambourine and kept 
in position by virtue of whirling 
motion 122 : on sun, moon, and stars, 
their nature and motion ibid. : moon 
nearest earth, stars next, and sun 
furthest away ibid. : explanation of 
tilt of earth's axis 122-3. 

' Little Astronomy,' The : a collection 
of astronomical treatises 317-20. 

λοξός κύκλος, the zodiac circle or eclip- 
tic 131. 

Lucretius 128-9. 

Macrobius 115, 131, 164,257, 258-9, 
311,340-1. 

Manitius 9 n., 312. 

Marinus of Tyre 346. 

Martianus Capella 113, 256, 314. 

Martin, Τ. H. 10 n„ 13-14, 16 n., 41, 
48 n., 101, 104, 105, 115,117, 119, 
141, 150, 154, 161-2, 167, 177, 178, 
185, 186, 188, 194 n., 219, 220 n., 
235 n -> 238, 250, 258, 270, 277. 

Maspero 48 n. 

Menaechmus, pupil of Eudoxus, 193, 
212. 

Menedemus 252. 

Menelaus, Sphaerica 318, 321. 

Menestratus 291. 

Menge, H. 325, 326 n. 



Meteoric stone, supposed prediction of 
fall of, by Anaxagoras 246. 

Meton : on length of seasons 200, 213, 
215 : Great Year of 19 years (iwea- 
καιδεκαετηρίς) or Meton's cycle 293-5, 
296 : observation of summer solstice 
in 432 B.C., 294. 

Metrodorus of Chios, on infinity of 
worlds 126-7. 

Milky Way: views on, of Parmenides 
67, 70-2, yy, Anaxagoras 83-5 and 
Democritus 83-5, 124-5, Pytha- 
goreans 118, 133, Oenopides 133: 
another view (' reflection ') 247-8 : 
Aristotle on, ibid. : by some con- 
nected with Plato's ' straight light ' 
150-1. 

Month : Greek month lunar 284 : 
1 hollow ' and ' full ' months 287, 
probably in use before Solon 285 : 
popular month of 30 days 285-6 : 
Solon's reform 285, 291 : intercala- 
tions of months 286, 287, 288, 293, 
296. 

Moon: substance and light of, 82, 124^. : 
gets its light from the sun (disco- 
very due to Anaxagoras) 19, 76, yj y 
78-9, 91, 158 : phases of, 60, 80, 120 : 
apparent diameter of, 313, 413 {see 
also under Sun) : estimates of size 
Hl-12,332,337,338, 340-1,342, 349, 
350 : estimates of distance 28, 37-8, 
1 14-15, 335-6, 338-9,